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APPLETONS’ MATHEMATICAL SERIES, 


PRACTICAL ARITHMETIC, 


BY 


G. P. QUACKENBOS, LL. D., 


AUTHOR OF 


“4 PRIMARY ARITHMETIC ;” “AN ELEMENTARY ARITHMETIC;” “AN ENGLISH GRAM- — 
MAR;” “FIRST LESSONS IN COMPOSITION;” “ADVANCED COURSE OF COMPO- 
SITION AND RHETORIO;” “A NATURAL PHILOSOPHY ;” “ILLUSTRATED 
SCHOOL HISTORY OF THE UNITED STATES;” “PRIMARY HIS- 
TORY OF THE UNITED STATES;” ETO. 


TPON THE BASIS OF THE WORKS OF 


GEO, R. PERKINS, LL. D. 


NEW YORK: 
D. APPLETON & COMPANY, 
649 & 351 BROADWAY. 
1873 


By the same Author: 


A PRIMARY ARITHMETIC: Handsomely Illustrated. 16mo, pp. 108. 30 cts. 

AN ELEMENTARY ARITHMETIC: 12mo, pp. 144. 50 cts. 

A MENTAL ARITHMETIC: Designed to impart readiness in mental calcula- 
tions, and extending them to the various operations needed in business 
life. 16mo, pp. 168. 45 cts. 

KEY TO PRACTICAL ARITHMETIC: 12mo, pp. 72. 20 cts. 

FIRST BOOK IN GRAMMAR: 16mo, pp. 120. 50 cts. 

AN ENGLISH GRAMMAR: 12mo, pp. 288. 90 cts. 

FIRST LESSONS IN COMPOSITION: In which the Principles of the Art are 
developed in connection with the Principles of Grammar. 12mo, pp. 182. 
90 cts. ; a 

ADVANCED COURSE OF COMPOSITION AND RHETORIC: A Series of 
Practical Lessons on the Origin, History, and Peculiarities of the English 
Language, Punctuation, Taste, the Pleasures of the Imagination, Figures- 
Style and its Essential Properties, Criticism, and the various Departments 
of Prose and Poetical Composition: Illustrated with copious Exercises. 
12mo, pp. 450. $1.50. 

ELEMENTARY HISTORY OF THE UNITED STATES: With numerous 
Illustrations and Maps, 12mo, pp. 216. '%5 cts. 

ILLUSTRATED SCHOOL HISTORY OF THE UNITED STATES. En- 
bracing a full Account of the Aborigines, Biographical Notices or Distin- 
guished Men, numerous Maps, Plans of Battle-fields, and Pictorial Mlustra- 

_ tions. 12mo, pp. 588. $1.75. 

A NATURAL PHILOSOPHY: Embracing the most recent Discoveries in 
Physics. Adapted to use with or without Apparatus, and accompanied with 
Practical Exercises and 335 Illustrations. New Edition, revised and brought 
in all respects up to date. 12mo, pp. 450. $1.75. 


ENTERED, according to Act of Congress, in the year 1866, by 
D. APPLETON & COMPANY, 


in the Clerk’s Office of the District Court of the United States, for the Southern 
District of New York, 


y 
/ 


-— 


N 


PREFACE. 


—_—~—— 


Tue Third of our Series of Arithmetics, designed for all ordi- 
nary classes in our Public and Private Schools, is now presented 
to the public. The aim has been to make it comprehensive, clear, 
free from verbiage in its definitions and explanations, inductive in 
its development of the subject, and well adapted to the school- 
room. 


It is believed that the study of Arithmetic, apart from its neces- 


~~ e 5 e e 

© sity as a practical branch, may be rendered invaluable as a mental 
“discipline. Every device has been resorted to in this work to 
~ make it_useful as a means of intellectual training, of teaching the 


te Ta a ea =e Se cree 
oung learner nd reason, at the same time without re- 
quiring anythin is not fairly within hisreach. Acting on this 


principle, the author has not laid down rules arbitrarily, but shown 


' the reasons for them by means of preliminary analyses. He has 


also placed occasional questions or suggestions after examples, in 
the belief that such hints, starting the learner in the right direc- 
tion, would encourage him to attempt the solution for himself, 
rather than apply for aid to his teacher,—a practice as destructive 
of self-reliance in the one as it is annoying to the other. 

To impress principles on the mind, as well as to impart facility 
in operating, much practice is necessary ; and, to secure this, 


~ humerous examples are presented, applying the rules in a great 


variety of ways. The answers in most cases are given, but, to 
test the learner, a few under almost every rule are omitted. 
Answers are apt to suggest the processes used; and, if they are 
invariably given, even the most faithful will unconsciously fall 


282261 


iv PREFACE, 


into the habit of depending upon them. <A Key for the teacher’s 
use will prevent any inconvenience at recitation. 

A “ Practical” Arithmetic should deserve its name, and we 
have kept this in view throughout. We have asked, What appli- 
cations of Arithmetic is the pupil likely to need in life? What 
are the shortest methods, and those actually used by business 
men? The branches of Mercantile Arithmetic have received 
|special attention,—the making out of bills, the casting of interest, 
partial payments, operations in profit and loss, averaging accounts, 
equation of payments, &c. Much collateral information on busi- 
ness subjects has been embodied. In a word, the author has 
weighed every line, with the view of giving what would be most 
useful and best prepare the learner for the duties of the counting- 
room. f 

The great distinguishing feature of this book is that it is adapted 
to the present state of things. The last five years have been five 
years of financial changes; specie payments have been suspended, 
prices have doubled, the tariff has been altered, a national tax 
levied, &c. No Arithmetic that ignores these changes should be 
placed in the hands of our youth. Time is too precious to be | 
wasted in learning things wrong, only to unlearn them on enter- 
ing into active life. Our examples are adapted to the present: 
the prices given are those of to-day; the difference between gold 
and currency is recognized and taught; the rates of duties agree 
with the present tariff; the mode of computing the national in- 
come tax is explained; a full description is given of the different 
classes of United States securities, with examples to show the 
comparative results of investments in them. These are matters 
that children, as well as adults, ought to know and understand. 

It is hoped that these, with other features that will be obvious 
on examination but need not be mentioned here, may commend 
the work to teachers generally. 


New Yor, August 10, 1866. 


CONTENTS. 


Cnartcr I, NUMBERS, . 
Cuap. If. NoTartIon, 

Arabic Notation, é . 

Roman Notation, 5 - 
CuHap. III. NUMERATION, . 
UCHAP.. IV... ADDITION, =... 
CHAP. VY. SUBTRACTION, ’ 
Cuar. VI. MULTIPLICATION, 


Cuap. VII. Division, 

Short Division, . 

Long Division, 

Relations of Dividend, re 
and Quotient, : 

Prime and Composite ene 
bers, “ 2 c F * 

Prime Factors, . A ‘ 

Cancellation, . , A 6 


CHap. VIII. Greatest COMMON 


DIvisor, : c 
Cuap. IX. Least Common MUr- 
TIPLE, : E 2 : 3 


Cuap. X. Common FRACTIONS, 
Reduction of Fractions, 
Addition of Fractions, 
Subtraction of Fractions, . 
Multiplication of Fractions, 
Division of Fractions, . 5 
Complex Fractions, : ; 


CHap. XI. DEciIMAL FRACTIONS, 
Notation of Decimals, 


168 


——_>—_— 
PAGE 
7% Addition of Decimals, - 
8 Subtraction of Decimals, 
8 Multiplication of Decimals, 
13 Division of Decimals, 
16 Reduction of Decimals, 
Circulating Decimals, ; 
t9 CHap XII. FEDERAL MoNEY, 
30 Making out Bills, . . 
38 | Cuap. XIII. Repuction, . 
48 Reduction Descending, . 
51 Reduction Ascending, . 
53 Reduction of Federal Money, 
Compound Numbers, . 
63 Sterling Money, ..." >. 
Troy Weight, . é 
64 Apothecaries’ Weight, 
65 Avoirdupois Weight, 
67 Long Measure, ‘ . 
Surveyor’s Measure, . " 
69 Square Measure, . 
Cubic Measure, : 
Liquid Measure, 
7 Beer Measure, 
4 Dry Measure, 
"9 Time, 
86 Circular Measure, 
88 Paper, . : : A 
90 Collections of Uhits, 
95 Reduction of Denominate Frac- 
95 tions, Common and Decimal, 159 
103 | CuHApr. XIV. Compounp ADDITION, 
- 103 | Coar. XV. Compound SUBTRAC- 
106 TION, A ° 6 ° 


Numeration of Decimals, 


meg 


Vi 
r PAGE 
CHap. XVI. Compound MuLTI!- 
PLICATION, . ; - 1% 
Cuap. XVII. Compounn Division, 179 
Practice, . 186 
Cuap. XVIII. Duopsctmats, . 188 
Cuap. XIX. PERCENTAGE, 194 
Profit and Loss, . A . 200 
CHap. XX. INTEREST, ; 205 
To find the Interest, . - 206 
To find the Rate, . . - 215 
To find the Time, . : 216 
To find the Principal, . 217 
Compound Interest, . 219 


CHap. XXI. NorTres.—PARTIAL 
PAYMENTS,—ANNUAL INTER- 

EST, 5 A s - ©2283 

Partial agmentat . ° 


U.S, Rules: + F « 225 
Mercantile Rule, 2 229 
Connecticut Rule, . 5 - 2380 
Notes with interest annually, 232 
CuAp. XXII.. Discount, ‘ . 233 
Present Worth, . ape. eet 
True Discount, . . 234 
Bank Discount, . 236 


, CHAP, XXIII. Commiss1on.—BRo- 


KERAGE.—STocks, . . 240 
Account of Sales, - 45 
Stocks, . 6 247 

CHap, XXIV. BANKRUPTCY, . 255 
CHap. XXV. INSURANCE,. . 256 
Accident Insurance, . : ae) 
Life Insurance, : é ee 209 
CHap. XXVI. Taxezs, . 4 . 260 
Assessment of Taxes, . eel 
National Tax, . 5 ; - 263 
Cuap. XXVIII. Dutms, .; . 264 


» 


CONTENTS. 


PAGE 


Cuap. XXVIII. "setae oF Pay- 


MENTS, . 5 4 - 266 
Averaging Accounts, . - 210 
Cash Balance, . ; A - 22 

CHap. XXIX. Ratio, A - 274 


CHap. XXX. PROPORTION, . - 276 
Simple Proportion, : att 6 
Compound Proportion, . 279 

Cuap. XXXI. ANALYSIS, . Bo posh! 
Reduction of Currencies, . « 285 
Colonial Currencies, . - «285 
Foreign Currencies, . . . 286 

Cuap. XXXII. ExcHAnes, oT ee8G 
Domestic Exchange, eae 
Foreign Exchange, - « aa 
Arbitration of Exchange, » 292 

CHap. XXXIII. PARTNERSHIP,. 294 

Cuap. XXXIV. ALLIGATION, ». 298 
Alligation Medial,. . . 298 
Alligation Alternate, rere) 

CHAP. XXXY. INVOLUTION, . 302 

CHap. XXXVI. EvyoLurTion, - 803 
Square Root, .  . - 3804 


Applications of Square Root, - 307 
Cube Root. . . 309 
Applications of Cube Root, - 812 
CHAP. XXXVII. PROGRESSION, 313 
Arithmetical Progression, . 313 
Geometrical Progression, . 316 


CHAP. XXXVIII.—MENSURATION, 318 


Parallelograms, . . . 318 
Drigncies 7 aaeure ° - . 318 
Circles? Fo xeemeae” hen 2. ED 
Cylinders; 7s 2.00 ueeeeete mits te eU 
Spheres, .. + eases ase Oee 
Cuap, XXXIX. ANNUITIES, . oce 
Cuar. XL. Tue Metric System, 329 
. 930 


Miscellaneous Examples, . 


PRACTICAL ARITHMETIC. 


CHAPTER I. 
NUMBERS. 


1, One, a single thing, is called a Unit, 

2. If we join another unit to onE, we have Two; if 
another, THREE; and so, adding a unit each time, we get 
FOUR, FIVE, SIX, SEVEN, EIGHT, NINE. 

3. One, two, three, &c., are called Numbers. <A 
Number is, therefore, one unit or more. 


4, Arithmetic treats of Numbers. 


5. Numbers are either Abstract or Concrete, They are 
Abstract, when not applied to any particular thing; as, 
one, eight. They are Concrete, when applied to -particu- 
lar things; as, one pound, eighé dollars. 


6. That to which a concrete number is applied, is called 
its Denomination. In the last example, dollars is the 
denomination of the number eight. 


7. Counting is naming the numbers in order; as, one, 
two, three, four, five, &e. 


8. We may express numbers by writing out their 
names, as one, two, three; or by characters, as 1, 2, 3. 


Quzstions.—1. What is a single thing called ?—2, What do we get by successive 
additions of a unit to one?—3. What are one, two, three, &c., called? Whatisa 
Number?—4. Of what does Arithmetic treat ?—5. How are numbers distinguished ? 
When are they called Abstract? When, Concrete ?—6, What is meant by the De- 
nomination of a concrete number?—7. What is Counting ?—S, How may we exprees 
numbers ? 


8 NOTATION, 


CHAP Tain 
NOTATION. 


9, Notation is the art of expressing numbers by char- 
acters. 
10. Two systems of notation are used, the Ar’abic and 
the Roman. 


The Arabic Notation. 


11. The Arabic Notation is so called because it was 
introduced into Europe by the Arabs, who obtained it 
from India. It uses ten characters, called Figures :— 


O i 2 3B 4 o G @ tc) 9 


NAUGHT ONE TWO THREE FOUR FIVE S1X SEVEN EIGHT NINE 


12, The first of these figures, 0, is called _Naught, 
Cipher, or Zero. It implies the absence of number. 

The other nine are called Significant Fi igures, or 
Digits,—each signifying a certain number. 


13. The greatest number that can be expressed with 
one figure is nine, 9. For numbers above nine, we com- 
bine two or more figures. 


First, 1 is placed at the left of each of the ten figures, forming 10, ten; 
ie eleven ; 12, twelve; 13, thirteen: 14, fourteen ; 15, fifteen , 16, six: 
teen ; 17, seventeen ; 18, eighteen ; 19, nineteen. 

Then 2, forming 20, twenty ; "91, twenty-one; 22, twenty-two; 23, 
twenty-three; 24, twenty-four; 25, twenty-five; 26, twenty-six; 27, 
twenty-seven; 28, twenty-eight ; 29, twenty-nine. 

Then 3, forming 30 (thirty), 31, 32, 33, 34, 35, 36, 87, 38, 39. 

Then 4, forming 40 (forty), 41, &c. Then 5; 50 (fifty), 51, &c. 
Then 6: 60 (sixty), 61, &. Then 7: 70 (seventy), 71, &c. Then 8: 
80 (eighty), 81, &c. Then 9: 90 (ninety), 91, &c. 


9. What is Notation ?-10. How many systems of notation are used? What are 
they called ?—11. Why is the Arabic Notation so calied? How many characters does 
ft use? What are they ?—12. What is the first of these figures called? What does 
itimply? What are the other nine called? Why are they called Significant ?— 
18. What is the greatest number that can be expressed with one figure? How do we 
express numbers above nine? Show how 1, 2. 3, &c., are combined in turn with each 
ef the ten figures, and what numbers are thus formed. 


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, oe cf : ; 2 ha 


THE ARABIC NOTATION. 9 


14, Units, Tens, Hunpreps.—The first or right-hand 
place is called the units’ place; the second, the tens’ place, 


1 in the units’ place ( 1) is 1 unit. 
1 in the tens’ place (10) is 1 ten, or ten units. 
2 in the tens’ place (20) is 2 tens, or twenty units. 
3 in the tens’ place (30) is 8 tens, or thirty units. 
A figure, therefore, in the second place denotes so many 
tens, and its value is ten times as great as if it stood in 
the are place. 


15. The value ofa figure standing alone or in the first 
place is called its Simple Value. Its value in any other 
place is called its Local Value. 


16. The greatest number that can be er pried with 
two figures is ninety-nine, 99. Next comes one hundred— 
100—expressed by putting 1 in the third place, which is 
called the hundreds’ place. 

To express hundreds, write the several figures in the 
third place with naughts after them :— 


One hundred, 100. _ Three hundred, — 300. 
Two hundred, 200. Four hundred, 400, &c. 
17. Observe how the numbers between the hundreds 
are expressed :— 
One hundred and one, 101—1 hundred, 0 tens, 1 unit. 
One hundred and two, 102—41 hundred, 0 tens, 2 units, &c. 
One hundred and ten, 110—41 hundred, 1 ten, 0 units. 
One hundred and eleven, 111 — 1 hundred, 1 ten, 1 unit, &c. 
Two hundred and one, 201 — 2 hundreds, 0 tens, 1 unit. 
Three hundred and one, 3801 —8 hundreds, 0 tens, 1 unit, &c. 
18, Figures are grouped in Periods of three each. 
These three places,. units, tens, hundreds, form the first 
period, or Period of Units. 


14. What is the first or right-hand place called? The second? What is the 
value of 1 in the tens’ place? 2 in the tens’ place? 35 in the tens’ place? What 
does a figure in the second place denote?—15, What is meant by the Simple Value 
of a figure, and what by its Local Value? Which of these always remains the same ? 
—16. What is the greatest number that can be expressed with two figures? What 
comes after 99? How is one hundred expressed? How are the hundreds expressed ? 
—17. Show by examples how the numbers between the hundreds are expressed.— 
18. How are figures grouped? What is the first period called? Of what three 
places does it consist ? 


1* 


eer re. iF 


10 


NOTATION. 


EXEROISE 


IN NOTATION. 


Express in figures, remembering that vacant places must be 
filled with naughts :— 


—s 


. Five units. 
. Three hundreds. 
. Eight tens, nine units. 


COsT SD OOF SS bo 


Four tens. 


. Six hundreds, four tens. 
Two hundreds, two units, 
. Seven hundreds, five tens. 
. 1 hundred, 1 ten, 5 units. 
. 1 hundred, 6 tens, 1 unit. 


. Five hundred and sixty. 

. Eighty-three. Thirty-nine. 
. One hundred and thirteen. 
. Nine hundred and nine. 

3. Seven hundred and fifty. 

. Two bundred and twelve. 
. Four hundred and eighty- 


seyen. 


19. Tuousanps :—The second period is that of Thou- 
sands, It consists of three places,—thousands, ten-thou- 
sands, hundred-thousands,. 

Observe how thousands are expressed :— 


One thousand, 1,000. Ten thousand, 10,000. 
Two thousand, 2,000. Fifty thousand, 50,000. 
Three thousand, 3,000, &c. Sixty thousand, 60,000, &c. 


One hundred thousand, 100,000. 
Seven hundred thousand, 700,000. 
Eight hundred thousand, 800,000, &c. 


20. Zo express a given number of thousands, write the 
number in the second period. If there are numbers cor- 


responding to the places of the first period, write them 
there ; if not, supply naughts. 


ExampiE 1.—Write seven hundred and nine thousand. 

To do this, write seven hundred and nine (709) in the second period. 
Supply three naughts for the units’ period—709,000. 

ExaAMpLE 2,—Write seven hundred and nine thousand, 
and forty. 


To do this, write 709, as before, in the second period, and 40 in the 
first, supplying a naught for the vacant hundreds’ place—709,040, 


19. What is the second period? Of what three places does it consist? Show 
how thousands are expressed.—20, Recite the rule for expressing a given number 
of thousands.—How do you write seven hundred and nine thonsand? Seven hun. 
dred and nine thousand, and forty ? 


THE ARABIC NOTATION. 11 
So, five hundred and fifty-one thousand, 551,000. 
Ten thousand, six hundred and eighteen, 10,618. 


Four hundred and sixty thousand, nine hundred, 460,900. 


EXEROISE IN NOTATION. 


Write the following numbers in figures :-— 

1. Fifty thousand. Four hundred thousand. 

2. Two thousand, two hundred and twelve. 

3. Two hundred thousand, six hundred and sixty-one. 

4, Eight hundred and twenty thousand, and thirty. 

5. Nine thousand, three hundred and seventy-one. 

6. Forty-seven thousand, one hundred and nineteen. 

7. Eighty-one thousand, and seven. 

8. Sixty thousand, four hundred and eighty-two. 

9. Seven hundred and twenty-eight thousand, eight hundred 
and fifty-seven. Ne 


21. Mizzions, Bittions, Trinuions, &c.—The third 
period is that of Millions, It consists of three places,— 
millions, ten-millions, hundred-millions, 


Exampies,—One million, 1,000,000. 
Ten million, 10,000,000. 
One hundred million, 100,000,000, 


22. The fourth period is that of Billions. It consists 
of three places,—billions, ten-billions, hundred-billions. 


Exampies.—One billion, 1,000,000,000. 
Ten billion, 10,000,000,000. 
One hundred billion, 100,000,000,000. 


23. The Periods above billions are seldom used. They 
are called Trillions, Quadrillions, Quintillions, Sextillions, 
Septillions, Octillions, Nonillions, Decillions, &e. 

Beginning at the right, name first the periods in order, 
then the places, as shown in the following Table:— 


21. What is the third period? Of what three places does it consist ?—-22. What 
is the fourth period? Of what three places does it consist? Give examples of the 
mode of expressing millions and billions,—23. Name the periods above billions. 
How many places must we fill, te express a million? To express a billion? ‘To ex- 
press ten thousand ? 


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6th Per. 5th Per. 


QUADRILLIONS TRILLIONS 


NOTATION. 


Hundred-billions, 
Ten-billions, 
Billions, 


-_—--—” 
4th Per 


BILLIONS 


24, One ten is equivalent 


to ten tens. 


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3d Per. 2d Per. 1st Per. 


MILLIONS THOUSANDS UNITS 


to ten units; one hundred, 


Hence, removing a figure one place to the 


right, diminishes its value ten times; removing it one 
place to the left, increases it ten times. 


25. Rute ror Noration.— Write in each period, be- 
ginning with the highest mentioned, the number belonging 
to it, filling vacant places on the right with naughts. 


The left-hand period need not contain three places, but every other 
must.—Naughts before a number do not affect its value, and should not 


be written. 


Every naught placed after a number throws its figures one 


place farther to the left, and therefore increases its value ten times. 
A person counting 100 every minute since the birth of Christ would 
not yet have reached a trillion. 


EXEROISE IN NOTATION. 


Write the following numbers in figures, placing units under 
units, tens under tens, &c.:— 
Nine hundred and eighty-one million, seven hundred. 
Ninety-six billion, one hundred million, and twelve. 

Three hundred and twenty quadrillion, five thousand. 
Fifteen quintillion, four quadrillion, ten thousand. 

Eight trillion, twelve billion, seven hundred. 

Two hundred and fifty-seven million, one hundred and 
ninety-one thousand, seven hundred and sixty-three. 


1B 


5, 
6. 


24. What is the effect of removing a figure one place to the right? Of remoying 
it one place to the left?—25. Give the rule for Notation. Which of the periods must 


sontain three places, and which need not? 
toa namber? Of annexing a cipher? What remark is made, to show how great a 


trillion is? 


What is the effect of prefixing a cipher 


- ia a ia ee. At ats TARTS 


EXERCISE IN NOTATION. 13 


7, Ninety-eight sextillion, three hundred million, eleven thou- 
sand, four hundred and thirteen. 

8. Seven hundred and seven trillion, forty-one million, seven 
hundred and twenty thousand, and one. 

9, Four hundred and fifty trillion, five hundred and forty bil 
lion, forty-five million, fifty-four thousand, and eleven. 

10. 475 decillion, 200 nonillion, 84 octillion, 7 septillion, 63 
sextillion, 450 quintillion, 2 trillion, three hundred and. sixty. 

11. Eight hundred and ninety-one quadrillion, one trillion, 
fifty billion, six hundred and nine million, and seventy. 

12. Two nonillion, fourteen septillion, two hundred and eleven 
quadrillion, thirteen trillion, five hundred and forty-six billion, 
twenty-seven thousand, and ninety-five. 

13, Twenty quintillion, two hundred and seven Dillion, six 
hundred million, six thousand, and fifty-nine. 

14. Two hundred sextillion, and sixty-nine. 

15. One trillion, one hundred billion, and eleven. 


The Roman Notation. 


26. The Roman Notation, so called because it was 
used by the ancient Romans, employs seven letters. 
I. denotes one; V., five; X., ten; L., fifty; C., one hun- 
dred; D., five hundred; M., one thousand. 


V resembles the outline of the hand with the five fingers spread. X is 
two V’s, or fives, Joined at their points, C begins the Latin word centwm, 
one hundred. It was sometimes written in this form £, and the lower 
half, afterwards written as L, denoted fifty. M begins the Latin word 
mille, a thousand. A thousand was sometimes written CIO; hence ID, 

_ afterwards changed to D, denoted 500.—Some think that V was used to 
denote five, because U, which was anciently written V, was the fifth vowel. 


27. These letters are combined to express numbers, 
according to the following principles :— 

1. Ifa letter is repeated, its value is uEve sated. XX. is 
twenty; Til. is three. 


26. Why is the Roman Notation so called? What does it use to represent num- 
bers? Why is it supposed that V was used for 5, and X for 10? How did C come 
to denote 100, and L 50? Why was M used for 1000, and D for 500?—27. In com- 
bining these characters to express numbers, what is the effect of repeating a letter? 


14 


NOTATION. 


2. A letter of less value, placed after one of greater, 


unites its value to that of the latter. 


VI. is-six. 


3. A letter of less value, placed before one of greater, 


takes its value from that of the latter. 


IV. is four. 


4, A letter of less value, placed between two of greater, 


takes its value from that of the other two united. 


is fifty-four. 
BoA bar over a letter increases its value a thousand 


LIV. 


times. V. is five thousand. 
ASB EAE 
I. One. L. Fifty. 
Ik. Two. LX. Sixty. 
II. Three. LXX. Seventy. 
iVi L OCT, LXXX. Eighty. 
V. Five. XC. Ninety. 
SIA Rtaywre, ©. One hundred. 
VII. Seven. CI. One hundred and one. 
VU. Eight. CO. Two hundred. 
IX. Nine. OCC. Three hundred. 
Xe bens CCCC. Four hundred. 
XI. Eleven. D. Five hundred. 
XII. Twelve. DC. Six hundred. 
XI. Thirteen. DCC. Seven hundred. 
XIV. Fourteen. DCCC. Eight hundred. 
XV. Fifteen. DCCCC. Nine hundred. 
XVI. Sixteen. M. One thousand. 
XVII. Seventeen. MM. Two thousand. 
XVIII. Eighteen. MMM. Three thousand. 
XIX. Nineteen. MMMM. Four thousand. 
XX. Twenty. V. Five thousand. 
XXI. Twenty-one. X. Ten thousand. 
XXX. Thirty. L. Fifty thousand. 
XL. Forty. M. One million. 


What is the effect of placing a letter of less value after one of greater? Of 
placing a letter of less value before one of greater? Of placing one of less value 
between two of greater? Of placing a bar over a letter? ‘Learn the Table. 


THE ROMAN NOTATION. 15 


The Roman Notation is now used chiefly in expressing dates, marking 
the hours on clock and watch faces, paging prefaces, and numbering vol- 
umes, chapters, or lessons of books. It was ill adapted for use in calcu- 
lating or keeping accounts, and was for most purposes superseded with 
great advantage, in the 16th century, by the Arabic Notation, which had 
been introduced among the learned of Europe two hundred years before, 
and had been gradually made known by means of almanacs, : 


EXERCISE IN NOTATION. 


Write the following numbers, first by the Arabic, and then by 
the Roman, Notation :— 


1. Eighteen. 5. Seventy-nine. 
2. Forty-five. 6. Eight hnndred. 
3. Six hundred. 7. Ten thousand. 
4, Three thousand. 8. Twenty-nine, 


9. Fifteen hundred (or, one thousand, five hundred). 
10. Nineteen hundred and five. 
11. Twelve hundred and thirty-eight. 
12. One million, one thousand, and one. 
13. Five thousand, seven hundred and ninety. 
_ 14. One hundred thousand, and eleven. 
15. Fifty thousand, four hundred and tifty-four. 
16. Sixteen hundred and ninety-nine. 
17. One thousand, six hundred and sixty: 
18. Two thousand, two hundred and eighty-seven. 


Express the following numbers according to the Roman Nota- 
tion: 825; 138; 10,500; 81; 119; 50,909; 1,000,000; 48; 5,555; 
76; 1,864; 4,200; 14; 15,000; 849; 1,111; 52; 660; 101,000. 

Express the following numbers according to the Arabic Nota- 
tion! “MoeoXxXV. MDOOO. LXVII.° VV. - LOCO? XVI 


LI. Ae COOCEY,) tO VT. AU ar TX OR 
MDCCOCLXXXIX. OXIX. 


ive fli: exe ets ee Sex vil. Ixxsecmiyss exliv:. Ixxxk 


For what is the Roman Notation now chiefly used? When was it superseded 
by the Arabic? When was the Arabic Notation first introduced, and how was it 
made known? 


16 NUMERATION. 


CHAPTER III. 
NUMERATION. 


28. Numeration is the art of reading numbers expressed 
by characters. 


29. Rute For Numeration.—l. Beginning at the 
right, divide the number into periods of three figures each. 

2. Beginning at the left, read the figures in each period 
as if they stood alone, adding the name of the period in 
every case except the last. 


The figures of the right hand period are never named as units, the word 
- units being understood. We read 500 jive hundred, not jive hundred units. 
Places containing 0 must be passed over in reading. We read 2043 

two thousand and forty-three, not two thousand, no hundred and Sorty-three. 


EXAMPLES, 


1,100 One thousand, one hundred; or, eleven hundred. 
140,000 One hundred and forty thousand. 
20,009,000 Twenty million, nine thousand. 
11,000,000,017 Eleven billion, and seventeen. 

9,000,070,212,005 Nine trillion, seventy million, two hundred and 

twelve thousand, and five. 
6,020,100,009,000,400 Six quadrillion, twenty trillion, one hundred bil- 

lion, nine million, four hundred. 


EXEROISE IN NUMERATION. 


Name the places in order—units, tens, hundreds, &c.; then 
read the number :— . 


ite 840000 7. 3'7123000000415863 
2. 75819 8. 714629300000927000 
3. 30451000 9 46327723207003 
A, 1673549000227 ° 10. 15615000000111101 
5. 84001100206 Ii. 827716420018 
6. 290902029092209 12. %423065056506650128 


28. What is Numeration ?—29. Give the rule for Numeration. Which period 
has its name understood in reading ? How do we read places containing 0? Give 
an example. 


NUMERATION. shai 


18.  4987048799080465 91. TGV 
14, 224000000600317010 99. MOXCII. 
15. 563745119 23. MXDOCCL. 
16. 1612875962 94, DOCOXIX. 

6 18459223 25. OXVIII. 
18. DRLEX: 26. MCCLXXIV. 
19. CCCCXCVIIL. 27. LMCXXII. 
20. DCOCCLXXXIL. 28.  WCCCXXXIII. 


29. Fill nine periods’with ones. Read this number. 

30. Fill seven periods with fours. Read this number. 

31. Set down 7 ten-thousands, 1 thousand, 5 hundreds, 6 tens, 
2 units. Read the number thus formed. | 

32. Set down 2 trillions, 3 ten-billions, 8 ten-millions, 6 mil- 
lions, 1 thousand, 9 tens, 5 units, filling vacant places with naughts. 
Read this number. 


30. Enatiso Numeration Tasre.—The division of 
numbers into periods of three figures each, as shown on 
page 12, is that followed in the United States, France, 
and the continent of Europe generally. The English 
divide into periods of stx figures each, naming them and 
the places they contain as follows :— 


n a a 

& a iS) 

eae = eae 

a —s ow ort 

58 BB HS 

sag ce a cag 

nH OS nin € ng v3 

Cc Yn cam Pe SO Los} “4 @ Lo] 

Ae oo a Gar of qe ag | 

Ey Ouyaaes. 3s Om gq 3s 0 °S 8 Si 

aa Bn so an Ax a of 

oUCUH os 5 Ue a 5 UH S ae! 

PP eel ego ee ees ee ee 

na lamer wm) 2) st aaa th & a3, (Sh ont. 

ro ar ree A ethers) S Meh a a Oe 

oeogmg@ar gZ oa = o a8 =. 2 eo RS 

os ee see Sak Se he gas Ba EB Sees 

SS Bee So oe ee Ae eS Se Se 
Ci . | ° 

als 3 eo ees SA pee a a Sue a oa Sis 8 a 

DFR RR RR HHA RRR BReERRRAR Gap 

= me He HB BH 
el (See ey a Se (Oe et ee 


4th Per. 3d Per. 2d Per. 1st Per. 
TRILLIONS BILLIONS MILLIONS UNITS 


30. How does the English Numeration Table differ from ours? Name the first 
eighteen places accoraing to our table; according to the English table. When we 
speak of a billion in this country, how many million do we mean? 


18 ADDITION. 


CHAPTER IV. 
ADDITION. 


31, Four girls and three boys went a riding; -how 
many went in all ? 


Here we are required to find one number containing as ae units as 
4 and 3 together. This process is called Addition. 


32. Addition is the process of uniting two or more 
numbers in one, called their Sum, Adding 4 and 3, we 
have 7 for their sum. 


AppDITION TABLE, 


0 and 1 are 1; 0 and 2 are 2, 0 and any number make that number. 
1 and 0 are 1; 2 and 0 are 2; any number and 0 make that number. 


2 and. 3 and 4 and 5 and 
1 are 2 1 are 3 l are 4 l are 5 l are 6 
2 are 3 2are 4 2 are 5 2 are 6 2 are. 7 
3 are 4 3 are 5 3 are 6 3 are 7 3 are 8 
4 are 5 4 are 6 4 are 7 4 are 8 4 are 9 
5B are 6 5 are 4¥% 5 are 8 5 are 9 5 are 10 
6*are 7 6 are 8 6 are 9 6 are 10 6 are 11 
7 are 8 Tare 9 7 are 10 7 are 11 7 are 12 
8 are 9 8 are 10 8 are 11 8 are 12 8 are 13 
9 are 10 9 are ll 9 are 12 9 are 13 9 are 14 
10 are 11 10 are 12 10 are 18 10 are 14 10 are 15 


1 are 7 l are 8 lare 9 1 are 10 1 are 11 
2 are 8 2 are 9 2 are 10 2 are 11 2 are 12 
38 are 9 3 are 10 3 are ll 3 are 12 8 are 13 
4 are 10 4 are 11 4 are 12 4 are 13 4 are 14 
5 are ll 5 are 12 5 are 13 5 are 14 5 are 15 
6 are 12 6 are 13 6 are 14 6 are 15 6 are 16 
4 are 13 7 are 14 4 are 15 7 are 16 7 are 17 
8 are 14 8 are 15 8 are 16 8 are 17 8 are 18 
9 are 15 9 are 16 9 are 17 9 are 18 9 are 19 
10 10 10 


$1. In the given example, what are we required to do? What is this process 
called ?—32. What is Addition? Whatis the result of addition called? How much do 
0 andany number make? How much do any number and 0 make? Recite the Table. 


ADDITION. 19 


33. Addition is denoted by an erect cross +, called 
Plus, placed between the numbers to be added. 6-+5 is 
read six plus five, and means that six and five are to be 
added. 

34, Two short horizontal lines =, placed between two 
quantities or sets of quantities, denote that they are 
equal. 6+4=10 is read six plus four equals ten, and 
means that the sum of six and four is ten. 

35. Observe the following :— 


38 +2—=5 | 445=—9 3+7=—10 5+8=13. 
then then then then 
18+2=15 44+35=39 538+ 7=60 5+28=—338 
23 +2—=25 44+-45—49 63+7=10 45+ 8=—538 


838 4+2=35, &. | 44+55=59, &e. | 734+7=80,&e. | 5+4+88=93, &c. 


36. Observe that 4+5=—9,and5+4=9. 
Hence, when numbers are to be added, it makes no 
difference which is taken first. 


EXEROISE ON THE ADDITION TABLE. 


How many are 7 and 6? Gand 7? 17 and6? 16 and 7? 
6and 27? 6and47? 57 and6? Sand 5? 5 and 8? 

How many are5 and 7? 85 and 7? 87 and 5? 7 and 85? 
5 and 87? 4 and 8? 4 and 88? 884+1+8? .4+2? 10442? 
1644+2? 17442? 27442? 9244? 40244? 

How many are 9 and 3? 29 and 3? 8 and 2? 48 and 2? 
108 and 2? 102 and 8? 8 and 8? 88 and 3? 8 and 78? 
178 and 3? 278 and 8? 11 and 1? 15 and 2? 10 and 2? 
20and3? 5and 30? 6and3? 23 and 6? 

Add 9 and 7% 9 and 9. 9 and 8 9 and 5%. 9 and 59. 
9and 58. 9and 47. 9and5. 4,5, and9. 5, 4, and 19. 

Add 9 and 2. 6,8, and2. 89 and2. 2and29. 1, 2,and1. 
49 and1. 51,1, and8 9and161. 1 and 179. 

How is 6+4=10 read ?—35. How much is 84+2? 1842? 8342? 445? 4435? 


4445? 347? 534+7? 6347? 38477?—86. How much is 4+5? How much is 
5+4? What principle is deduced from this? 


20 ADDITION. 


What does 6+6equal? 6+106? 12646? 226+6? 6436? 
46+3+8? 564442? 54+1+46? 

What does 4+6 equal? 104+62 2446? 6414? 5+6? 
854+6? 5476? 94+4? 9+44? 

What is the sum of 10 and 10? 10 and 20? 10 and 380? 
4,6,and70? 7,8,and80? 10and5? 4,5,and8? 38,6,and19? 
4,2,and7? '7,3,and2? 8,4,and8? 21,5,and3? 98,4, and 3? 

Count by twos, commencing 2, 4, 6, 8, &¢., up to 100. 

Count by twos, commencing 1, 3, 5, 7, &c., up to 99. 

Count by threes, commencing 3, 6, 9, 12, &c., up to 99. 

Count by threes, commencing 2, 5, 8, 11, &., up to 98. 

Count by threes, commencing 1, 4, 7, 10, &c., up to 100. 

Count by fours, commencing 4, 8, 12, 16, &c., up to 100. 

Count by fives, commencing 5, 10, 15, 20, &c., up to 100. 


37, APPLICATIONS OF AppiT1I0onN.—To find a whole, | 


when its parts are given, add the parts. 

38. To find the selling price, when the cost and gain 
are given, add the cost and gain. 

39. To find the cost, when the selling price and loss 
are given, add the selling price and loss. 

40. To find a later date, when an earlier date (A. D., or 
after Christ) and the difference of years are given, add 
the earlier date and the difference of years. 


MENTAL EXEROISES. 


1. A boy has 20 cents in one pocket, 9 in another, and 3 ina 
third; how many cents has he in all? 

: Ans. 20+9+8 cents, or 82 cents. 

2. A farmer buys.a cow for $32*, and sells her so as to gain 
$7; what does she bring? (See § 38.) 


87. How do you find a whole, when its parts are given ?—88. How do you find 
the selling price, when the cost and gain are given?—39. How do you find the cost, 
when the selling price and loss are given ?—40. How do you find a later date, when 
an earlier date and the difference of years are given? 


* This mark ($) denotes dollars, It is always placed before the number. $32 is read thirty-two dollara. 


Ee _—— 


EXERCISE IN ADDITION. 21 


3. Sold a picture for $33, at a loss of $8; what did the picture 
cost? (See § 39.) ‘ 

4, In what year will Charles be eight years old, if he was 
born in 1861? (See § 40.) 

5. How many strokes will a clock strike in twelve hours, com- 
mencing at 1 o'clock? 

6. I spent $10 for a coat, $4 for a vest, and $6 for a hat; 
what did the whole cost? 

7. Five cows in one field, three in another, and seventeen in 
a third, make how many cows altogether ? 

8. A grocer has 10 barrels of flour, 5 of sugar, and 6 of pota- 
toes; how many barrels has he in all? 

9. Clay was born in 1777. Webster was born 5 years later; 
what was the date of Webster’s birth? 

10. There are 14 bones in the face, 6 in the ears, and 8 in the 
hack of the skull; how many bones in the whole head? 

11. Washington became president in 1789. He held the office 
eight years; when did he leave it? 

12. There are 31 days in July, and 31 in August; how many 
days in both months ? 


Mopet.—Thirty-one is 8 tens and 1 unit. 38 tens and 1 unit, added to 3 tens and 
1 unit, make 6 tens and 2 units, or 62. Ans. 62 days. 


13. A person travelled 40 miles by railroad, and 35 miles by 
stage; how far did he go in all? 

14. Genesis contains fifty chapters, and Exodus forty; how 
many chapters in both? 

15. I have three lots of land; the first contains 30 acres, the 
‘ second 10, and the third as many as the other two together. 
How many acres in all three? 

16. A certain orchard contains 16 apple-trees that bear, and 
4 that do not; 7 pear-trees that bear, and 3 that do not; and 10 
cherry-trees. How many apple-trees does it contain? How 
many pear-trees? How many trees altogether ? 

17. If a person spends $10 on Monday, $5 on Tuesday, and as 
much on Wednesday as on both the previous days, how many 
dollars does he spend altegether? 


Oe ADDITION. 


41, Appine sy Cotumns.—When the numbers are 
too large to be added mentally, we write them down and 
add the columns separately. 

ExamPLe.—A merchant gives $5261 for one lot of 
goods, $432 for another, and $303 for a third. How much 
do they all cost him ? 


_They cost him the sum of $5261, $489, and $303. That we may 
unite things of the same kind, in writing the numbers 
down, we place units under units, tens under tens, Xe. 


Begin to add at the bottom of the right-hand column. $5261 
_3 units and 2 units are 5 units, and 1 are 6 units ; 432 
write 6 under the units. 0 tens and 8 tens are 8 tens, and 303 


6 are 9 tens; write 9 under the tens. 8 hundreds and 4 
hundreds are 7 hundreds, and 2 are 9 hundreds; write Ans, $5996 
9 under the hundreds. Bring down the 5 thousands, Ans. 

$5996. 


42. Proor or Appition.—Proving an example is 
finding whether the work is correct. 

Addition may be proved by adding the columns from 
the top downward. If the sum is the same as when they 
are added from the bottom upward, we infer that the work 
ts right. If an error has been made in the first addition, 
it is not likely to be repeated in the second, when the 
numbers are taken in a different order. 


ExampLe.—Prove the example in $41. Add each Toh ea 
column from the top downward. 1 and 2 are 3, and 3 are 

6. 6and3 are 9. 2 and4 are 6, and 38are 9. Bring down 303, 
5. Ans. $5996,—the same as before. Hence the work is 

Brioni. Ans. $5996 


EXAMPLES FOR PRAOTIOE. 
Read and add the following numbers; prove each example :— 


(1) (2) (8) 
62317 1100264 29761005483147382 
4330 53105 47211574162 
FIAS2T 58234510 22851040001004 


41. How do we deal with the numbers, when they are too large to be added men- 
tally? In the given example, how must we write down the numbers to be added ? 
Why so? Proceed with the addition.—42, What is meant by proving an example? 
How is addition proved? Why should the result be the same when you add in the 
opposite direction ? (§ 36) Why is not an error in the first addition likely to be re- 
peated in the second? Prove the example in § 41. 


EXAMPLES IN ADDITION. "23 


4, Add 901538, 321, 405301, and 3214. Ans. 498989. 
5. Add 84600325, 70, 55402, and 123201. Ans. 84778998. 
6. Add 41, 725, 12, 200, 4001, 20, and 3000. Ans. 7999. 
7. Add 11, 232, 2543, 14201, 870012. Ans. 886999. 
8. What is the sum of 1640263, 1501, 214123, and 23011? 
9. What is the value of 26+231041+711+55101+2110? 


10. Add thirteen hundred; nineteen million, two hundred 
thousand, five hundred; forty-two; five hundred and twenty-four 
thousand, and thirteen; and twenty million, fourteen thousand, 
and thirty-two. Ans. 89739887. 

11. Add eleven million, two hundred and twenty-three thou- 
sand, four hundred and fifty-one; five hundred and ten thousand, 
two hundred and fifteen; five million, one hundred and forty-one 
thousand, one hundred and twenty-two; and twelve thousand, 
two hundred. Ans. 16886988. 

12. What is the sum of 14 billion, three hundred and twenty- 
one; 2 billion, 15 million, 111 thousand, three hundred and five ; 
420 million, 12 thousand, and fifty-three; and 131 million, 600 
thousand ? Ans. 16566723679. 

13. Find the sum of twenty trillion, two hundred billion, two 
million, and seventeen; thirty-one billion, three hundred and 
seventy-one million, six hundred and thirty-four thousand; thir- 
teen thousand, three hundred and twenty-one; and five billion, 
eleven million, twenty-one thousand, four hundred and forty. 

Ans, 20236384668778. - 

14, Add eleven million and eleven; five million, two hundred 
and ninety-two thousand, one hundred and twenty-three; and 
six thousand, five hundred and fifty. Ans. 16298684. 

15. One town contains 16735 inhabitants, another 22242; 
what is the population of both? 

16. How many acres in three farms, if the first contains 427 
acres, the second 250 acres, the third 211 acres ? 

17. A person who is worth $145250, makes $10000 more, and 
has $220700 left to him. What is he then worth ? 

18. If an army of 23452 men is reénforced with 15316 men, 
how many will it then contain? 


94 ADDITION. 


19. A boat starts with 1652 bushels of wheat aboard; 48 
miles down the river, it receives 27 barrels of flour and 8385 
bushels of wheat. How many bushels of wheat are then aboard ? 

Ans. 4987 bushels.” 


43. The sum of a column may consist of more than 
one figure. In this case, place the right- -hand figure 
under the column added, One add the left-hand figure or 
Jigures to the next left-hand column. 


If the sum of a column is 64, write down 4 and add 6 to the figures 
of the next left-hand column ; if it is 127, write down 7 and add 12 to 
the next column. 


Exampie.—Add 3658, 4903, 7006, and 734. 


Write the numbers, units under units, tens under 
tens, &c. Begin to add at the right. The sum of the 


units is 21 ,—2 tens and 1 unit. Write the 1 unit in the 3 Pris 
units’ place, and add the 2 tens to the tens in the next eee 
column. 2 pga 
2 and 8 are 5,and 5 are 10. 10 tens are 1 hundred seres 
and 0 tens. Write 0 in the tens’ place, and add the 1 3658 
hundred to the hundreds in the next column. 4903 
1 and % are 8, and 9 are 17, and 6 are 23. 23 hun- 
dreds are 2 thousands and 3 hundreds. Write 3 in the 7006 
hundreds’ place, and add the 2 thousands to the thou- 734 


sands in the next column. 
2, and 7 are 9, and 4 are 13, and 3 are 16. 16 thou- Ans. 16301 

sands are 1 ten-thousand and 6 thousands. This being 

the last column, write them down. <Azs. 16301. 


EXAMPLES FOR PRAOTICE. 
Read and add the following numbers; prove each example :— 


(20) (21) (22) 
49778 857215 24313755596 
112 29524 24464485 
352243 8461489 5325273374 
5314 ; 828 306482265 
51544382 58327317 3694817601153 
65423216 874516526 12365498705618 


43. When the sum of a column consists of more than one figure, what must we 
do? Ifthe sum ofa column is 64, what do we do? Ifitis98? Ifitis127? Ifthe 
sum of a column is expressed by three figures, how many do we write down, and how 
many do we add to the next column? Add the numbers in the given example. 


ADDITION. 25 
44, Rule for Addition, 


1. Write units under units, tens under tens, &e. 

2. Beginning atthe right, find the sum of each column. 

3. If the sum is expressed by one figure, write i 
under the column added ; if not, place the right-hand fig- 
ure there, and add the left-hand figure or figures to the 
next column. 

4, Prove by adding in the opposite direction. 


EXAMPLES FOR PRAOTIOE. 


. Add 123405, 2354210, 354, 794327, and 36547. 

. Add 27562, 345607, 2461, 4567801, and 365. 

. Add 1034001, 78954, 3'79205, 367001, and 45637. 
. Add 11, 4562, 773, 15266, 8958, and 66666. 

5. Add 1003875, 406780, 4673005, 4112, 18365791, 2478, and 
164857. Ans. 23716898. 

6. What is the sum of three hundred thousand, six hundred 
and fifty; seven thousand, eight hundred and thirty-two; eleven 
thousand, five hundred and sixty-seven; ten thousand and fifty- 
six; four hundred and seventy-two? Ans. 330577. 

7. A man*drew five loads of bricks. In the first load, he 
had 1209; in the second, 1453; in the third, 1101; in the fourth, 
1212; in the fifth, 1803. How many bricks were there in all ? 

8. If there are shipped from the United States, 15624 barrels 
of flour to Sweden, 250 barrels to Holland, 205154 to England, 
6401 to Cuba, and 19602 to Mexico, how many barrels are 
shipped altogether ? 

9. Find the sum of eighty-eight million, and nineteen; forty- 
seven thousand, and sixty-eight; nine million, seven hundred and 
eighty-five thousand; nine hundred and eighty-six; eight billion, 
seven million. Ans. 8104833078. 

10. How many square miles are in the British Isles, there 
being 50922 square miles in England, 30685 in Scotland, seven 
thousand three hundred and ninety-eight in Wales, and in 
2 


Hm CF DO ee 


26 ADDITION. 


Ireland eighteen hundred and twenty-seven more than in Scot- 
land ? Ans. 121517 square miles, 

11. It is computed that there are two million pagans in North 
America, two million in South America, one million in Europe, 
five hundred and ten million in Asia, sixty-five million in Africa, 
and twenty-four million in Oceania. How many pagans are there 
in the world? Ans. 604,000,000. 

12. Maine, the largest of the New England States, contains 
31766 square miles. New York, the largest of the Middle States, 
contains 15234 square miles more than Maine. How many square 
miles in New York? 

13. A person has $1557 in one bank, $2348 in another, and 
in a third as much as in both the other two. How much has he 
in the third bank? How much in all three? 

14. A lady gave $3445 for a house and $1055 for furniture. 
She then bought some adjoining land for as much as both house 
and furniture cost. What did she give for the whole? 

15. Wellington’s army at Waterloo consisted of 49608 in- 
fantry, 12402 cavalry, and 5645 artillery-men. How many men 
did it contain in all? 

16. Napoleon’s army at Waterloo consisted of 48950 infantry, 
15765 cavalry, and 7232 artillery-men. How many men did it 
contain in all? ‘ 

17. How many men did both Wellington’s and Napoleon’s 
army at Waterloo contain ? Ans. 139602 men. 

18. How many bushels of wheat are there on four boats, each 
of which contains 5250 bushels of wheat and 45 barrels of flour? 

Ans. 21000 bushels, 

19. President Madison was born in 1751, and attained the age 
of eighty-five; in what year did he die? 

20. How many strokes does a clock strike in 24 hours? 

21. A lady gave each of her three daughters $9250, and her 
son $8345. How much did she distribute among them ? 

22. The earth is 95298260 miles from the sun. The planet 
Neptune is 2767105740 miles farther. What is Neptune’s distance 
from the sun? 


ADDITION. oF 


23. A has $4250; B has $875 more than A; C has as much | 
as A-and B together. What are all three worth? Ans. $17750. 

24. How far is it from New York to Buffalo, the distance from 
New York to Albany being 150 miles, from Albany to Rochester 
251 miles, and from Rochester to Buffalo 75 miles? 

25. A man worth $12500 makes as much more, and has $5490 
left to him. What is he then worth? Ans. $80490. 

26. Required the whole population of the world, that of N. 
America being estimated at 46000000; S. America, 20000000; 
Europe, 280000000; Asia, 680000000; Africa, 80000000; and 
Oceania, 28000000. 

27. How many men in an army consisting of four regiments, 
two of nine hundred and eighty men each, and two of twelve 
hundred and forty ? Ans. 4440 men. 

28. A merchant bought $1786 worth of books, and $875 worth 
of stationery. On the books he gained $549, and on the station- 
ery $228. What did he sell the books for? What did he sell 
the stationery for? What was his whole gain? 

Ans. Books, $2835. Stationery, $1103. Gain, $777. 

29. In 1862, the postal revenue of the U. S. amounted to 
$8299820; in 1863, it was $2863969 more. What did it then 
amount to? 

30. If I invest $2356 in pork, and $1977 in beef, and sell 
them so as to gain $395, how much do I receive for the whole? 

81. A man left his wife $95000; each of his three sons, $15000 ; 
his daughter, $34000; and the rest of his property, which 
amounted to $47250, to charitable societies. What was the 
whole value of his estate? Ans. $221250. 

32. A’s orchard contains 146 apple-trees; B’s, 22 pear-trees, 
9 plum-trees, and 27 apple-trees; O’s, 18 plum-trees, 189 apple- 
trees, and 88 pear-trees. How many apple-trees in all three 
orchards? How many pear-trees? How many plum-trees? 
How many trees altogether ? 

33. The salary of the Vice-President of the United States is 
$10000 a year; that of the President is $40000 more. What 
de the yearly salaries of both amount to? 


28 ADDITION, 


34. A certain school opens with 78 boys and 129 girls, 
Within 80 days there is an addition of 42 boys and 89 girls. How 
many does the school then contain? 

35. Add LXVI., MDXIX., CCIV., XVIII. Ans. 1807. 

86. Add MD., VOXXX., XLIV., CXV., X. Ans. 1015789. 


45. Practise the following examples till they can be 
added at sight up and down, naming the results only. 
Thus, in Example 1 :—three, eight, fourteen, sixteen, 
seventeen, twenty-three, thirty, thirty-four, thirty-six— 
write 6 in the units’ place, and add 3 to the next column. 


(1) 


1179582 28204681 76456789 5234567 
2295344 17130579 76789123 6346789 
3381437 96792468 13123456 2678912 
4574296 85246835 63456789 7891235 
5275011 74683579 55789123 1124567 
6443322 63357924 21123456 4456789 
7109876 52642753 44456789 4678913 
8123345 31297531 32789123 8892345 
93451238 97468864 76123456 8123456 
(5) (6) (7) (8) 
7389234 66694375 34751212 7634725 
6188345 53693025 23586259 8583614 
5267345 35215354 11310344 9472583 
1856234 86424443 97311581 8361472 
2945245 14644636 80678363 7258361 
3134123 53301445 72846256 6747258 
4723345 49321435 62562172 5136147 
2312234 21648673 57166249 4825836 
5689345 36623535 47691554 3614725 
6490133 55615525 34426235 2583614 
3567345 41623573 20934389 9472583 
1289234 24635521 192812138 8361472 
7631405 36159247 42816354 7258361 


(2) 


(8) 


(4) 


ADDITION. 29 


46. When the sum of a column consists of three fig- 
ures, the two left-hand figures must be added to the next 
column. Thus, in Ex. 9, the sum of the first column is 
108—write down 8, and add 10 to the next column. 


@) (10) (il) (12) 
846750 68362T 635643 802950 
846750 562961 428744 395135 
846750 963421 935613 801915 
846759 137653 453824 717486 
846759 412623 254872 496395 
846759 525636 662816 585289 
846759 5472138 654883 674198 
846759 325443 544865 717487 
846759 202652 111834 496396 
846759 517650 653851 585285 
846759 531021 452840 674199 
846759 446133 232890 717488 
846759 107553 623886 496397 
846759 547605 651893 585286 
353759 838714 734882 674195 

12208358 7349899 8033336 9420101 
(18) (14) (15) 
1399736957 3826486095 7587897897 
7228226934 6667483794 4182596897 
6179552483 988253592 5469675797 
5598504587 8759236294 4829827957 
4499246912 486755 2789494849 
3398715975 7631486745 4589296895 
4297751987 4782884793 9286795997 
5181736661 3292692 1743394797 
6548424326 5884276195 6949417875 
7464352987 85486785 7489386822 
6394756980 8926386411 3589095897 
5298656957 6867141394 9791297897 
4164157256 3788445795 2278096895 
3547399992 4949951678 7639155256 


71201220994 


63161299018 


78215431728 


30 


SUBTRACTION. — 


CHAPTER. V. 


SUBTRACTION. 


47, Five hens are ona roost. Three fly down; how 
many remain ? | 
Here we are required to take 3 from 5, or to find the difference between 


3 and 5. This process is called Subtraction. 


48, Subtraction is the process of taking one number 


from another. 


SUBTRACTION TABLE. 


0 from 1 leaves 1; 0 from 2, 2; 0 from any number leaves that number. 


1 from 
1 leaves 0 
2 leaves 1 
3 leaves 2 
4 leaves 3 
5 leaves 4 
6 leaves 5 
7 leaves 6 
8 leaves 7 
9 leaves 8 

10 leaves 9 


6 from 
6 leaves 0 
7 leaves 1 
8 leaves 2 
9 Teaves 3 

10 leaves 4 
11 leaves 5 
12 leaves 6 
13 leaves 7 
14 leaves 8 
15 leaves 9 


2 from 
2 leaves 0 
3 leaves 1 
4 leaves 2 
5 leaves 3 
6 leaves 4 
7 leaves 5 
8 leaves 6 
9 leaves 7 

10 leaves 8 
11 leaves 9 


7 from 
7 leaves 0 
8 leaves 1 
9 leaves 2 
10 leaves 3 
11 leaves 4 
12 leaves 5 
13 leaves 6 
14 leaves 7 
15 leaves 8 
16 leaves 9 


3 from 
3 leaves 0 
4 leaves 1 
5 leaves 2 
6 leaves 3 
7 leaves 4 
8 leaves 5 
9 leaves 6 

10 leaves 7 
11 leaves 8 
12 leaves 9 


A from 
4 leaves 0 
5 leaves 1 
6 leaves 2 
4 leaves 8 
8 leaves 4 
9 leaves 5 

10 leaves 6 
11 leaves 7 
12 leaves 8 
13 leaves 9 


5 from 
5 leaves 0 
6 leaves 1 
7 leaves 2 
8 leaves 3 
9 leaves 4 

10 leaves 5 
11 leaves 6 
12 leaves 7 
13 leaves 8 
14 leaves 9 


8 from 
8 leaves 0 
9 leaves 1 

10 leaves 2 
11 leaves 3 
12 leaves 4 
13 leaves 5 
14 leaves 6 
15 leaves 7 
16 leaves 8 
17 leaves 9 


9 from 
9 leaves 0 


10 leaves 1. 


11 leaves 2 
12 leaves 3 
13 leaves 4 
14 leaves 5 
15 leaves 6 
16 leaves 7 
17 leaves 8 
18 leaves 9 


10 from 
10 leaves 0 
11 leaves 1 
12 leaves 2 
13 leaves 3 
14 leaves 4 
15 leaves 5 
16 leaves 6 
17 leaves 7 
18 leaves 8 
19 leaves 9 


47. Repeat the example. What are we here required to do?—48, What is Sub 


- yaction? What does 0 from any number leave? Recite the Table. 


SUBTRACTION. ; ol 


49, The number to be subtracted, is called the Subtra- 
hend; that from which it is to be taken, the Minuend. 
The result is called the Remainder, or Difference. 


3 from 5 leaves 2; 3 is the subtrahend, 5 the minuend, 2 the re- 
mainder or difference.—If the minuend is less than the subtrahend, the 
subtraction can not be performed; we can not take 3 from 2. 

50. Subtraction is denoted by ashort horizontal line —, 
called Minus, placed before the subtrahend. 5—3 is 
read five minus three, and means that 3 ts to be subtracted 
Jrom 5. 


51. Observe the following — 


3—2=1 7 —3—4 Y= 4 11—10=1 
then then then then 
13—2=—11 47—3—44 27 —23—4 31—10=21 
23 —2=21 57—3—54 1 W—i3—4 51—10=—41 


33—2—31,&c.| 67—38=—64,&ce.}| 87—83=4, &c.| 91—10=8], &e. 


EXERCISE ON THE SUBTRAOTION TABLE. 


Subtract 4 from 5. 4 frem 15. 14 from 15. 4 from 24. 
4 from 44. 54 from 55. 2 from 6. 2 from 66. 62 from 66. 

Take 3 from 5. 8 from 75. 3from85. 83 from85. 1 from 9. 
1 from 19. 11 from19. 3 from 8. 8 from 38. 

How much is 1—1? 3-3? 238—28? 88-232 48—33? 
43—10? 53—10? 54-10? 6-32 86—82 86—832 86—10? 
8—4? 28—4? 28-24? 9-5? 49-5? 9—72 69-7? 

Take 3 from 9. 3 from 59. 3 from 99. 2 from 10. 2 from 
20. 2from 60. 6from 10. 6 from 50. 6 from 80. 

‘How much is 12-7? 29-7? 42-7? 92-7? 15-8? 
65—8? "75-82? 14-9? 384-9? 44-9? 64-9? 15-10? 
25-10% 16-92? 26-9? 76—9? 86-9? 10-4? 10-8? 


49. What is the number to be subtracted called? Whatis the number from 
which the subtrahend is to be taken called? What is the result called? 8 from 5 
leaves 2; select the minuend, subtrahend, and remainder. In what case can the sub- 
traction not be performed ?—50. How is subtraction denoted? How is 5—8 read? 
What does it mean ?—51. How much is 83—2? What follows? How much is 7—3? 
27—23? TT—T3? How muchisl1—10? 81—10? 51—10? 


32 SUBTRACTION. 


Subtract 8 from 14. 8 from 64. 8 from 84. 6 from 13 
6 from 83. %from14. 7 from 74. Sfrom17. 8 from 87. 
Take 7 from 10. 9from18. 6from15. 2 from39. 8 from 
47, 4from56. 25 from 29. 36 from 88. 7% from 11. 
Count backward by twos from 100. Thus: 100, 98, 96, &e. 
Count backward by twos from 99. Thus: 99, 97, 95, &e. 
Count backward by threes from 99. Thus: 99, 96, 93, &c. 
Count backward by fours from 100. Thus: 100, 96, 92, &c. 
Count backward by fives from 100. Thus: 100, 95, 90, &e. 


52, AppiicaTions oF Susrraction.—When a whole 
and one of its parts are given, to find the other part, 
subtract the given part from the whole. 

53, When a whole and all its parts but one are given, 
to find that one, subtract the sum of the given parts from 
the whole. 

54, When the cost and selling price are given, to find 
the gain, subtract the cost from the selling price. 

55. When the cost and loss are given, to find the sell- 
ing price, subtract the loss from the cost. 

56. When the selling price and gain are given, to find 
the cost, subtract the gain from the selling price. 

57. When a later date and the difference of years are 
given, to find an earlier date (4. D., or after Christ), sub- 
tract the difference of years from the later date. 


MENTAL EXEROISES. 


1. A grocer who has 19 barrels of flour, sells 10 of them. 
How many has he left? Ans. 19—10, or 9, barrels. 

2. Leaving home with $17, I spend $5 and give $4 away. 
How much have I left? (See § 53.) 


52. When a whoie and one of its parts are given, how do we find the other part? 
—53. When a whole and all its parts but one are given, how do we find that one '— 
54. When the cost and selling price are given, how do we find the gain?—5d. When 
the cost and loss are given, how do we find the selling price ?—56. When the selling 
price and gain are given, how do we find the cost?—57. When a later date and the 
difference of years are given, how do we find an earlier date? 


MENTAL EXERCISES. 338 


8. A colt was bought for $81, and sold for $88. What was 
the gain? (See § 54.) 

4, A butcher lost. $7 on a cow that cost $49. What did he 
sell her for? (See § 55.) 

5. A jeweller sold a ring for $29, and thereby gained $3, 
What did the ring cost him? (See § 56.) 

6. La Fayette was born in 1757. President Madison was born 
six years earlier. What year was that? (See § 57.) 

7. If ten gallons of wine are drawn out of a hogshead con- 
taining 63 gallons, how many are left? 

8. I sold a watch for $57, and by so doing gained $5. How 
much did it cost? 

9. Napoleon died in 1821. When was the battle of Waterloo 
fought, which took place six years before his death? . 

10. A farmer who has 89 sheep, sells 52 of them. How many 


does he retain ? 


Movpet.—Eighty-nine is 8 tens and 9 units; fifty-two is 5 tens and2 units, 5 
tens and 2 units from 8 tens and 9 units leave 3 tens and 7 units, or 87. Ans, 87 
sheep. 


11. If I buy some cloth for $95 and sell it at a loss of $32, 

what do I get for it? 
12. A person lays out $4 for books, $2 for paper, and $1 for 

pens. Uow much change must he receive for a $20 dollar bill? - 

13. A boy who has 58 cents, gives 82 cents tothe poor. How 
many cents has he left? 

14. Ifa man buys a cow for $45 and a calf for $6, and seils 
both for $62, how much does he make by the operation ? 


58, When the numbers are too large to perform the 
operation mentally, write the smaller number under the 
greater and subtract each figure from the one above it. 

ExampLte.—A person who has $87 ee gives away 
$6035. How much has he left ? 

He has the difference between $6035 and $87945, which is to be found 
by subtraction, Write the smaller number under the greater,—units under 


58. When the numbers are too large to perform the operation mentally, how do ~ 
we “aal with them? Go through the given example, 


34 SUBTRACTION. 


units, tens under tens, &c., because things can be taken only from others 
of the same kind. 
Begin to subtract at the right. 5 units from 4... 

5 aie leave 0 units ; write 0 in the units’ Méinwend $87945 
column. 8 tens from 4 tens leave 1 ten; write Swbtrahend 6035 
it down. O hundreds from 9 hundreds leave 9 f oa es 
hundreds. 6 thousands from 7 thousands leave Remainder $81910 
1 thousand. Bring down 8. Ans. $81910. 


59. Proor or Susprracrion.—Add the remainder and 
subtrahend. If their sum equals the minuend, the work 
is right.—This follows, because a whole is equal to the sum 
of its parts. The minuend is the whole; the remainder 
and subtrahend are its parts. 


. 681910 

Exampie.—Prove the above example. Add the re- Air 8 6035 
mainder and subtrahend. Their sum is $87945, which. SM ue sebcunnps 
equals the minuend. Hence the work is right. Sum $87945 


EXAMPLES FOR PRAOTICE. 


Read minuend, subtrahend, and remainder. Prove each ex- 
ample :— 


(1) (2) (3) 
From 8267054 93847765138 87945568746598 
Take 145031 624324122 52345154133273 


4, Subtract 61425626377889 from 573929699387989. 
5. From VDCCCLXXXIV. take MCCOCXLIII. 

6. How much more is excviii. than xxxvi.? 

7. From five hundred and sixty-three billion fifty-nine thou- 
sand and seven, take two hundred and twenty billion thirty-five 
thousand and four. Ans. 343000024003. 

8. Subtrahend, four billion five million and three; minuend, 
eighteen trillion seven billion nineteen million and six; required 
the remainder. Ans. 18003014000003. 

9. From sixty-eight million nine hundred thousand and six- 
teen, take seven million two hundred thousand and two. 


59. How is subtraction proved? Why must the sum of the remainder and sub- 
trahend equal the minuend? Prove the example in § 58. 


RULE FOR SUBTRACTION. 35 


60, The lower figure may represent a greater number 
than the one above it. 
ExamPLeE.—F rom 964 take 839. 


We can not take 9 units from 4 units. Hence from the 5 
6 tens we take 1, leaving 5 tens. 1 ten is equal to 10 
units, which we add to the 4 units, making 14. Now sub- 964 
tracting 9 units from 14 units, we have 5 units left, which 839 


we write in the remainder, in the units’ place. 38 tens from 
5 (not 6) tens leave 2 tens. 8 hundreds from 9 hundreds Ans, 125 
leave 1 hundred. -Ans. 125. 

When we come to the second column, in stead of taking 1 from the 
tens of the minuend, as was done above, we may add 1 
to the tens of the subtrahend. Thus: 9 from 14, 5. 1 964 
and 8 are 4; 4 from 6,2. 8 from 9,1. Ans. 125, the 839 
same as before. This method is more convenient, and the 
result must be correct, for the addition of 10 units to the 195 
minuend is counterbalanced by the addition of 1 ten to the Ans. 
subtrahend, 

By some, this adding of 10 to the upper figure is called borrowing, 
aud the adding of 1 to the next lower figure carrying. 


61. We may have to repeat the process just described, 
several times in succession. 
ExamMpLe.—From 980000 take 969893. 


8 from 10,7. 1 and 9 are 10; 10 from 10,0. 1 980000 
and 8 are 9; 9 from 10, 1. land 9 are 10; 10 from 969893 
10,0. 3 1, and,6 are. ; 7 from.8, 1.9. from, 9,0; Sie 
haughts at the left are not written. Avs. 10107. Ans, 10107 


62. Rule for Subtraction. 


1. Write the smaller number under the greater, units 
under units, tens under tens, &e. 

2. Beginning at the right, take each figure of the sub- 
trahend from the one above it, and write the remainder 
under the figure subtracted. 

3. If any lower figure is greater than the one above tt, 
add 10 to the upper figure; subtract, and add 1 to the 
next lower figure. 

4. Prove by adding remainder and subtrahend. 


60. From 964 take 839, explaining the several steps.—61. What is this adding of 
10 to the upper figure called by some? What is adding one to the next lower fig- 
ure called? Show, with the given example, how we may have to repeat this pro« 
tess several times in succession.—62. Recite the rule for subtraction. 


36 : SUBTRACTION. 


EXAMPLES FOR PRAOTIOE. 


Read the numbers. Subtract. Prove each example :— 


@) (2) (8) 
801732641 156241098755 743812634378021 
5162388264 64980099668 56424152883322 

(4) On hea (5) 
89900321098 5601312499324 4385768506870 
11127717749 999746446289 4039299991989 

7. From 248008 take 14652. 15. 8630145416—9218682. 

8. From 8146380 take 79999. 16. 245610085—81740305. 

9. Take 90648 from 300652. 17. 51849085 —22683518. 
10. Take 89989 from 89990. 18. 426834260—97958473. 
11. Take 42329 from 52330. 19. 98765432 —23456789. 


12. From 8264531 take 7642. 20. 10008674—10007987. 
13. Take 15623 from 824618, 21. 576301498—85600534. 
14, From 900061 take 10378. 22. 7000338251—531258. 

23. A merchant sells a lot of flour for $12085, and thereby 
gains $996; what did it cost him? (See § 56.) 

24. Victoria became queen in 1837, 771 years after the Nor- 
man Conquest. What was the date of the Conquest? (See $57.) 

25. From thirteen billion subtract 8621856, and from -their 
difference take the same subtrahend. Ans. 12982757288. 

26. The population of the United States in 1863 was estimated 
at 34,344,926; in 1862, at 33,344,589. What was the increase ? 

27. A warehouse containing goods valued at $295125 took 
fire. Only $27250 worth of goods was saved; what was the 
value of those consumed ? 

28. Georgia was first settled by Oglethorpe in 1738. How 
long was that after the settlement of Virginia at Jamestown, 
which took place in 1607? 

29. A man gave $21460 for a farm, and $1685 for stock. If 
he sold the whole for $25000, did be gain or lose, and how much? 


— 


EXAMPLES FOR PRACTICE. 37 


80. If a person who has 86048 bushels of wheat sells one lot 
of 1845 bushels, and another of 12067 bushels, how much has ‘he 
left ? Ans, 22131 bushels. 

31. O sells D 58 barrels of apples for $116, 225 watermelons 
for $30, and 100 chickens for $28. D pays $95 cash; how much 
does he still owe C? Ans. $79. 

32. A lady divided $10000 among her three children. - She 
gave the eldest $3585, and the second $3196. How much did 
the youngest receive? (See § 53.) 

83. At an election, 12847 votes were cast; the successful 
candidate received 8968; how many were for his opponent? 
What was the majority of the former ? Ans. Maj. 5089. 

34. A broker, at the end of a day’s business, had on hand 
$04253. How much of this was in bills, $14160 of it being in 
gold, and $1789 in silver ? 

35, A library contained 1429 volumes in English, 876 in 
French, and 198 in German. 642 of these books were burned 
up, and 183 sold; how many were left? Ans. 1178 vols. 

36. P and Q begin business with $4500 each. P gains $368; 
Q loses $419. Which is worth the most, and how much ? 

87. The Imperial Library of Paris contains 1000000 printed 
volumes and 84000 manuscripts. The Royal Library of Munich 
contains 800000 volumes and 18600 manuscripts. How many 
volumes in both libraries? How many manuscripts? How many 
more volumes than manuscripts? 

88. The native population of New York state in 1860 was 
2882095; that of Pennsylvania, 2475710. The foreign-born 
population of N. Y. was 998640; that of Pennsylvania, 480505. 
What was the population of both states? How many more native 
inhabitants in N. Y. than in Penn,? How many more inhab- 
itants of foreign birth? How many more inhabitants altogether? 

39. Add the difference between $6453 and 64987 to the dif- 
ference between 7000 and 5999. Ans, 22467, 

40. From the sum of 26348 and 14275, take their difference. 

41. What is the value of 18648 +270967—4689 ? 

42. From forty-one billion subtract 863246 + 579868. 


38 MULTIPLICATION, 


CHAPTER VI. 
MULTIPLICATION. 


63. What will 5 lemons cost, at 3 cents each ? 


If 1 lemon costs 3 cents, 5 lemons will cost 5 times 8 cents, or 18 
cents. Here we are required to take 3 five times. This process is called 
Multiplication. 


64. Multiplication is the process of taking one of twe 
numbers as many times as there are units in the other. 


Mourrerication Taste. 
Once 0 is 0; twice 0 is 0; 0 taken any number of times is 0. 
0 times 1 is 0; 0 times 2 is 0; 0 times any number is 0. 


Once Twice 3 times 4 times 5 times 6 times 
‘Nee ae | 123agsiae Las a3 1s 34. T xis 5 dis LG 
2 ig 2 OMisees 2is 6 2is 8 2 is 10 2 Wl 
313° 3 318.26 3 is 9 3 is~12 8 is 15 3 is 18 
4is 4 4is 8 4 is 12 4 is 16 4 is 20 4 ig 24 
5 is 5 5 is 10 5 is 15 5 is 20 5 is 25 5 is 80 
6is 6 6 is 12 6 is 18 6 is 24 |. 6.is 80 6 is 386 
718 of 4" is 14 7 is 21 7 is 28 Tivo 7 is. 42 
8 is 8 8 is 16 8 is 24 8 is 32 8 is 40 8 is 48 
9is 9 9 is 18 9 is 27 9 is 36 9 is 45 9 is 54 
10 is 10 | 10 is 20 | 10 is 30 10 is 40 | 10 is 50 | 10 is 60 
Theis 11-1)-11 4s 22:..|.11 Gs 8847 1}is: 44.11 As 55011 4s: 66 
Poss Sey 12 vas ed he so Gasiede is 46 12 is 60 12 49-72 
7 times 8 times 9 times 16 times | Li times | 12 times 
1 is. 4 lis 8 fis. .99 bo luistel0O wad eal T¥is Vk 
219 414. 2 18.16 9 sig 1 8-1, 6248-20 2 is. 22 2is 24 
Solseo: 3 iss. 3.518 “271. 4-on 18) 250 Chis 35. 3 18 .36 
4 is 28 4 is 32 4 is 36/] 4 is 40 4 is 44 4 is 48 
5 is 85 5 is 40 5 is 45| 5 is 50 51s 5d 5 is 60 
6 is 42 6 is 48 6 is’ 54+ 6 ts 60°} 6 is 66) Bis ue2 
4 is 49 7 is 56 Y.385°6839 4830490 ¢ et ast VEO Ae ae 84 
8 is 56 8 is 64 8.16 672 | Sits: 80). 84s, 88: Bais 196 
9 is 63 9 is 72 9 is 81 9 is 90 9is 99 9 is 108 
10 is 70 | 10 is 80 | 10 is 90] 10 is 100 | 10 is 110 | 10 is 120 
11 is 77 | 11 is 88 LL"'19s-99 441 4s 140 Fal ts See a as 132, 
12 is 84 12 is 96 12 is 108 | 12 is 120 | 12 is 182 | 12 ig 144 


MULTIPLICATION. 39 


65. The number to be taken, or multiplied, is called 
the Multiplicand. The number by which we multiply, 
or which shows how often the multiplicand is to be taken, 
is called the Multiplier. The result, or number obtained 
by multiplication, is called the Product. 

3 times 2 is 6. 2 is the multiplicand, 3 the multiplier, 6 the product. 


66. The multiplicand and multiplier are called Factors 
of the product. 2 and 3 are factors of 6. 

67. Multiplication is denoted by a slanting cross x, 
placed between the factors. 2x3 is read, and denotes, 
two multiplied by three. 

68. The multipler shows how many times the multi- 
plicand is to be taken. Multiplying 2 by 3 is taking 2 
three times: 2+2+2=6. 2x38=6. Multiplying is, 
therefore, a short way of adding a number to itself. 

69. When two numbers are to be multiplied together, 
it makes no difference in the result which is taken as the 
multiplicand, and which as the multiplier. 4 x 3= 12. 
3xXx4= 12. 


Here we have 12 stars, whether we take them crosswise * * * 
as forming 3 rows of 4 each, or lengthwise as forming 4 * * * 
rows of 3 each. Epa ucnne? 


* * * 


70. When the multiplier is an abstract number, the 
product is of the same denomination as the multiplicand. 
3 times 2 men is 6 men; 38 times 2 apples is 6 apples, &e. 

71, Apprication or Murtipiication.—W hen the num- 
ber of articles and the cost of one are given, multiply 
them together, to find the awhole cost. 


63. In the given example, what are we required todo? What is this process 
called ?—64, What is Multiplication? How much is 0 taken any number of times? 
How much is 0 times any number? Recite the Table-—65. What is the number to 
be multiplied called? What is the number by which we multiply called? What is 
the result called? Give an example of these definitions.—66, What are the multipli- 
cand and multiplier called? Name the factors of 6; of 10,—67. How is multiplica- 
tion denoted? How is 2x38 read?—68. What does the multiplier show? Give an 
example. Multiplying is, therefore, a short way of doing what ?—69. In multiplying 
two numbers, what is found to make no difference? Prove this.—70. When the mul- 
tiplier is an abstract number, of what denomination is the product ?—71. How do you 
find the cost of a number of articles, when the cost of one is giyen? 


40 MULTIPLICATION. 


MENTAL EXEROISES. 


How much is 8 times 6? 6 times 8? 9 times5? 5 times 9? 
Y times 10? 10 times 7? 11 times 5? 5 times 11? 

How much is 8x12? 6x6? 2x9? 4x7? 12x5? 1x6? . 
0x10? 1x10? 11x22? 10x38? 5x0? 9x6? 8x7? 9x9? 

What is the product of 10 and 11? 6 and 38? 12 and 10? 
M9and 8? lland11? 5and5? 12and12? 8 and 12? 

How much is 4x3x7? 2x5x8? 1x8x8? . 2x6x1? 
6x5x0? 8x3x8? 4x8x6?2 2x8x®x7? 

1. If 6 marbles are bought for 1 cent, how many can be 
bought for 4 cents? Ans. 4 times 6, or 24, marbles, 

2. 7 days make a week. How many days in 8 weeks? In 5 
weeks? In 7 weeks? In 11 weeks? 

8. A certain boy earns $3 a week. How much will he earn 
in4 weeks? In 6 weeks? In 9 weeks? 

4, At 10 cents apiece, what will 2 writing-books cost? 8 
writing-books? 12 writing-books? 

5. Two boys have four pair of ducks each. How many ducks 
have they in all? 

6. How many bushels of pears in four gardens, each contain- 
ing three trees, if each tree yields two bushels? 

7, A person having six broods of eleven chickens each, sells 
two of the broods. How many chickens has he left? 

8. If a boat goes 12 miles an hour, how far will it go in 2 
hours? In 5 hours? In 8 hours? 

9. Ifa boy reads 4 pages every morning and 5 every afternoon, 
how many pages will he read in 7 days? 

10. If a man earns $12 a week, and spends $7, how much will 
he lay up in 1 week? In 4 weeks? 


72, Mutriretyine By 12 or Less.—The Multiplication 
Table carries us as far as 12 times. We can, therefore, 
multiply by 12 or any less number in one line. 

Rute.— Write the multiplier under the multiplicand, 
units under units, tens under tens. Beginning at the 


MULTIPLICATION. 41 


right, multiply each figure of the multiplicand in turn, 
placing each product in the same column with the figure 
multiplied, 

ExAmMpPLe.—Multiply 53201 by 3. 


Writing 3 under the units’ figure of the mul- eS 
tiplicand, begin to multiply at the right. 8 times ultiplicand 53201 
lis 3; 8 times 0 is 0; 3 times 2is 6; 3times3 Multiplier 3 
is 9; 3 times 5 is 15. The last product consists <7 ae 
of two figures; set it down with its right-hand Product 159603 
figure under the figure multiplied. Ans. 159603. 


73. In the above example, each product except the 
last consists of but one figure. When any product con- 
sists of two or three figures, place the right-hand one 
under the figure multiplied, and add the rest to the next 
product. 


ExamMpLe.—Multiply 5309 by 12. 


Write the multiplier under the multiplicand, units 5309 
under units, tens under tens. Begin at the right. 

12 times 9 units are 108 units,—or, 10 tens and 8 12 
nits. Write the 8 units in the units’ place, and add the 
10 tens to the next product. Ans, 63708 


12 times 0 tens are 0 tens, and 10 are 10 tens,—or, 1 
hundred, 0 tens. Write 0 in the tens’ place; add 1 to the next product. 

12 times 3 hundreds are 36 hundreds, and 1 are 87 hundreds,—or, 
8 thousands, 7 hundreds. Write 7 in the hundreds’ place, and- add 3 
thousands to the next product. 

12 times 5 thousands are 60 thousands, and 3 are 68. This being the 
hst product, write down both figures. dns. 63708. 


i 
EXAMPLES FOR PRACTICE. 

(1) (2) (8) (4) 
Multiply 73214 832614 901432 20613 
By 2 3 4 5 

(5) (6) (7) (£) (9) 
802716 291503 237016 654321 61423 
6 7 8 9 10 


10. How much is 11 times 75992638401? Ans, 835919022411. 


72. With how great a multiplier can we multiply in oneline? Recite the rule. 
Tllustrate the process with an example.—73. When any product consists of more 
than one figure, how do we proceed? Illustrate the process with the given example. 


42 MULTIPLICATION, 


11. How much is 9 times 8608498751 ? Ans. 82476488759. 
12. What is the value of 847352619 x12? Ans. 10168231428. 


13. 372918635 x 2. 19. 7294880756 x 9. 

14, 896140753 x 1. 20. 960527834 x 10. 

15. 579063245 x 3. 21. 248781659 x 11. 

16. 472938619 x 5. 22. 581629473 x 12. 

17. 639845728 x 7. 23. 1759268423. x 4. 

18. 576838492 x 8. 24. 86738413895 x 11. 
Ob, Multiply 31486 by 2; 3; 4; 5; 6; 7; 8; 9; and add the 
products. Ee 1385384. 
26. Multiply 8976201 by 9; 8; 7; 6; 5; 4; 3; 2; and add 
the products. Ans. 394952844, 
27. Multiply 652081 first by 8, then by 8; and find the dif- 
ference between the products. Ans. 3260405. 
28. How much is twelve times six hundred and forty-nine 
thousand and thirty-seven? Ans. 7788444. 
29. Six is one factor, ninety-six thousand and seventy-three is 
the other. What is the product? Ans. 576438. 


30. What cost 1785 coats, at $11 each? 
31.. What is the product of CXCVIII. and XI.?. Ans. 2178. 
32. Multiply XDCCCXXXIV. by VIIL. Ans, 86672. 


74, MULTIPLYING BY NUMBERS ABOVE 12.—When the 
multiplier exceeds 12, multiply by its figures separately. 

ExampLe.—Multiply 287 by 156. 

We can not multiply by 156 at once. Hence we first 
multiply by the 6 units; then by the 5 tens, or 50; then 
by the 1 hundred. Thus we get three Partial Products, as 
they are called ; adding these, we get the whole product. 


Multiplicand 287 
Multiplier 156 


oe 1722 =28%x 6 
ae 1485 = 28Tx 50 


Products 


Preduct 44772 = 287 x 156 


PROOF.—RULE. 43 


Here, when we come to multiply by 5, we write the first figure of 
the partial product under the 5. This is because the 5 is 5 tens, or 50, 
0 being omitted on the right. So, when we multiply by 1 hundred, we 
place the first figure of the partial product under the 1, two naughts 
being omitted on the right. Take care, then, always to place the first figure 
of each partial product under the figure used in multiplying. 


75, Proor or Murtiprication.— Multiply the 156 
multiplier by the multiplicand. If this product 287 
agrees with the former one, the work ts right. “1092 

ExampLe.—Prove the last example. 1248 


Multiply the multiplier 156, by the multiplicand 287, The 312 
product is 44772, which agrees with the former one, Hence 4447 
the work is right. 


76. When two numbers are to be multiplied together, 
it is usual to take the one with the fewer figures for the 
multiplier. 


Sy 
EXAMPLES FOR PRAOTIOCE. ™ 


Find the product. Prove each example :— 


1. Multiply 268 by 157. 6. 2463 x 1857. 
2. Multiply 418 by 234. 7. 1974 x 9486. 
3. Multiply 537 by 856. 8. 2684 x 2631. 
4, Multiply 916 by 729. 9. 47685 x 8249, 
5. Multiply 846 by 4823. 10. 16853 x 62583. 


77, Rule for Multiplication. — 


1. Write the multiplier under the multiplicand, units 
under units, tens under tens, &e. 

2. If the multiplier is 12 or less, multiply by it each 
Jigure of the multiplicand in turn, beginning at the right ; 
set down the right-hand figure of each product, and add 
the remaining figure or figures, if any, to the next product. 


74, When the multiplier exceeds 12, what are we to do? Multiply 287 by 156, 
explaining the process. What are the products obtained by multiplying by the dif- 
ferent figures of the multiplier called? Where must we take care to write the first 
figure of each partial product? Why so ?—75. How is multiplication proved? Prove 
the last example.—76. When two numbers are to be multiplied together, which do 
we take for the multiplier ?—77. Recite the rule for Multiplication. 


44 MULTIPLICATION. 


3. If the multiplier exceeds 12, multiply by each of its 
Jigures in turn, writing the first figure of cach partial 
product under the figure used in multiplying. Then add 
the partial products. 

4, Prove by multiplying multiplier by multiplicand., 


78. Naueut in THE Moutrtietier.—When 0 occurs in 
- the multiplier, bring it down, and go on multiplying by 


the next figure, all in the same line. "967 
ExampiE.—Multiply 7967 by 4005. 4005 
First multiply by 5. Bring down the two naughis, 

each in its own column. Then multiply by 4, setting 39835 

the product in the same line; its first figure is thus 3186800 

brought under the 4. Finally, add the partial prod- smarts age ee 

ucts. Ans, 31907835 


EXAMPLES FOR PRAOTICE. 


1. Multiply four thousand two hundred and ninety-three, by 
eight thousand and seventy-six. Ans. 84670268. 

2. Multiply fifty-seven thousand and three, by seventy-five 
thousand and four. Ans. 4275453012. 

3. The factors of a certain product are 51, 4, 6, and 108. 
Required the product. Ans. 132192. 

4, How much money must be divided among 2065 persons, 
that each may have $908 ? 

5, A drover who had 967 head of cattle, bought 92 more, and 
then sold the whole for $63 apiece. How much did he receive? 

6. How many books in three rooms, furnished with four book- 
cases apiece, each case containing 108 books? 

7. What cost 825 horses at $145 apiece? 

8. What is the product of 9263 and 7603? 

9. What is the value of 68753 x 10408? 

10. Multiply MDLXV. by VIX. Ans. 7839085. 

11. Multiply LCCXLI. by XVII. Ans. 502761687. 


78. When 0 occurs in the multiplier, how must we proceed? Illustrate this with 
the given example 


NAUGHTS AT THE RIGHT. (45 


79, Naveuts ar THE RIGuT.—We learned in § 25 that 
every naught placed after a number increases its value 
ten times. Hence, to multiply by 10, 100, 1000, &e., 
annex as many naughts as are in the multiplier. 


76x10=760 76 x 100 = 7600 76 x 10000 = 760000 


80. Exampre.—Multiply 4200 by 40. 
Multiplying in the usual way, we find 


42/00 the product to be 168000. The result is 
40 the same as if we had multiplied the sig- - Be: 
___ __hificant figures together, and annexed the a 0 
168000 ine at the right of both factors. Ans, 168000 


When there are naughts at the right of either factor 
or both, multiply the other figures, and annex to their 
product as many naughts as are at the right of both 
factors. 


EXAMPLES FOR PRACTIOEX.. 


Find the value of the following :— 


1. 80632 x 10. 7. 28000 x 146. 
2. 42635 x 100. 8. 976 x 25000. 
8. 62 x 100000. 9, 15000 x 1500. 
4, 8541 x 1000. 10. 64700 x 89000. 
5. 6000 x 1000. 11. 839100 x 60000. 
6. 14000 x 10000. 12. 6287000 x 7800. 


13. One gold eagle is worth $20; how many dollars are 6500 
eagles worth? 

14. If 100 pounds make one hundred-weight, how many 
pounds in seventy-eight hundred-weight ? 

15. Light travels 192000 miles in a second; how many miles 
will it travel in 60 seconds? 

16. What will 140 miles of railroad cost, at an average of 
$42900 a mile? 

79. What is the effect of annexing a naught toa number? How, then, may we 
multiply by 10, 100, 1000, &c. ?—80. Multiply 4200 by 40 in the usual way. What 


other way of obtaining the result is there? When there are naughts at the right of 
éither factor or both, what is the shortest mode of proceeding ? 


46 MULTIPLICATION, 


17. If an army consume 840 barrels of provisions in one La 
how many will it consume in 1 year, or 365 days? 

18. If sound travels 1120 feet in a second, how far will it 
travel in 20 seconds? 

19, A farmer has 6 orchards, each containing 10 piles of 
apples. In each pile are 1200 apples; how many apples in all? 

20. Multiply 640 by 10; by 50; by 940: add the products. 

21. How much more is 6800 x 140 than 970 x 850 ? 


81. Muttietyinc py A Composite Numsber.—A Com- 
posite Number is the product of two or more factors 
greater than 1. 16 is a composite number, being equal 
to 8 x 2. 

When the multiplier is a composite number, we may 
either multiply by the whole at once, or by its factors in 
turn. The result will be the same. Multiplication by a 
composite number may, therefore, be proved by multiply- 
ing by its factors. 

Hxampie.—Multiply 93 by 24. 


24=6x4 or, 8x3 or, 12x2 


93 93 93 93 
24 6 , 8 12 
872 “BBS G44 1116 
186 4 3 2 
2232 2232 2232 2232 


What are the factors of 33—that is, what two numbers multiplied 
together produce 83? What are the factors of 108? 72? 44? 21? 
182? 3852? 99? 422? 654? 49? 121? 


EXAMPLES FOR PRACTIOE. 


In examples 1—6, first multiply by the whole multiplier; 
then prove the result by multiplying by its factors. 


1. What cost 63 firkins of butter, at $16 apiece? 


81. What is a Composite Number? How may we multiply by a composite 
number? How may multiplication by a composite number be proved? 


EXAMPLES FOR PRACTICE. 47 


2. What is the weight of 84 barrels of flour, averaging 196 
ponnds each ? 

8. How many hours in 865 days, there being twenty-four 
hours in one day ? 

4, If a person travels 96 miles a day, for 108 days, how far 
does he travel in all? 

5. What will 27 miles of plank road cost, at $4200 a mile? 

6. How many bushels of apples does an orchard of 107 trees 
yield, if each tree produces 12 bushels? 

7. What will 18 square miles of land cost, at $17 an acre, 


there being 640 acres in one square mile ? Ans. $141440. 
8. How many miles will a locomotive go in 7 days of 24 hours 
each, if it moves 29 miles an hour? Ans. 4872 miles. 


9. The earth moves in its orbit 68000 miles an hour. How 
far will it move in 865 days of 24 hours each? 

10. In an orchard of 219 apple-trees, the average yield of each 
tree was 8 barrels of fruit, worth $3 a barrel. What was the 
whole yield worth? Ans. $1971. 

11. A man owing $8213, gives in payment 5 horses valued at 
$175 each, 29.cows at $38 each, and $765 in cash. How much 
remains unpaid ? Ans. $5471. 

12. A ship, after sailing 23 hours east at the rate of 8 miles 
an hour, is driven west by a storm 10 miles an hour for 5 hours. 
At the end of the 28 hours, how far is she from where she 
started ? Ans, 134 miles. 

18. In one year there are 31556929 seconds. How much does 
one trillion exceed the seconds in. 1864 years? ‘ 

14, A man bought two farms; one of 87 acres, at $54 an 
acre; the other of 101 acres, at $37 an acre. He paid $8140; 
how much was still due? Ans. $295. 

15. If I give 8 horses, each worth $150, and 13 cows, each 
worth $56, for 50 acres of land, valued at $19 an acre, do I gain 
_ or lose, and how much? © Ans. Lose $228. 

16. If a man travels 47 miles a day for 5 days, and then goes 
54 miles a day for four days, how many more miles will he have 
to go, to complete a journey of 600 miles ? 


48 


DIVISION. 


CHAPTER VII. 
DIVISION. 


82. Two pints make a quart; how many quarts in 8 


pints ? 


If 2 pints make 1 quart, in 8 pints there are as many quarts as 2 is 
contained times in 8. Here we are required to find how many times 2 is 


contained in 8. 


This process is called Division. 


83. Division is the process of finding how many times 
one number is contained in another. 


2 in 3 in 4 in 5 in 6 in 7 in 

once. 8, once. 4, once. 5, once, 6, once. 7, once. 

4, twice. 6, twice. 8, twice. | 10, twice. 12, twice. | 14, twice. 
6, 3times.| 9, 8 times, | 12, 3 times. | 15, 8 times. | 18, 3 times. | 21, 3 times. 
8, 4 times. |12, 4 times. | 16, 4 times. | 20, 4 times. | 24, 4 times. | 28, 4 times. 
10, 5times. | 15, 5 times. | 20, 5 times. | 25, 5 times. | 30, 5 times, | 35, 5 times. 
12, 6 times, |18, 6 times. | 24, 6 times. | 30, 6 times. | 36, 6 times, | 42, 6 times. 
14, T times. |21, 7 times. | 28, 7 times. | 35, 7 times, | 42, 7 times. | 49, 7 times. 
16, 8 times. | 24, 8 times, | 32, 8 times. | 40, 8 times. | 48, 8 times. | 56, 8 times, 
18, 9 times. | 27, 9 times. | 36, 9 times. | 45, 9 times. | 54, 9 times. | 63, 9 times. 
20, 10 times. | 30, 10 times. | 40,10 times. | 50,10 times. | 60,10 times. | 70,10 times. 
22, 11 times. | 83, 11 times. | 44,11 times. | 55,11 times, | 66,11 times. | 77,11 times, 
24, 12 times. | 36, 12 times. | 48,12 times. | 60,12 times. | 72,12 times. | 84,12 times. 

8 in 9 in 10 in 11 in 12 in 
8, once. 9, once. 10, once 11, once. 12, once. 

16, twice. 18, twice. 20, twice. 22, twice. 24, twice. 
24, 3 times. 27, 3 times. 80, 3 times 83, 8times. | 36, 3 times. 
32, 4times. | 36, 4 times. 40, 4times. | 44, 4 times. 48, 4times. 
40, Stimes. | 45, Stimes. | 50, 5 times. 55, 5 times. 60, 5 times. 
48, 6 times. 54, 6 times. 60, 6 times. 66, 6 times. 72, 6 times. 
56, 7 times. 63, 7 times. 70, % times. TT, 7 times. 84, 7 times. 
64, 8 times. 72, 8 times. 80, 8 times. 88, 8 times. 96, 8 times. 
72, 9 times. 81, 9times. | 90, 9 times. 9, 9 times. | 108, 9 times. 
80, 10 times. | 90, 10 times. | 100, 10 times. | 110, 10 times. | 120, 10 times. 


88, 11 times. 
96, 12 times. 


———— 


82. Solve the given example. 


Division TABLE. 


Any number is contained in. 0, 0 times. 
1 in 1, once; 1 in 2, twice; 1 in 3, 3 times; 1 in 4, 4 times, &e. 


99, 11 times. 
108, 12 times. 


110, 11 times. 
120, 12 times. 


121, 11 times. 
182, 12 times. 


132, 11 times. 
144, 12 times. 


What are we here required todo? What is this 
process called ?—83, What is Division? How many times is any number contained 
in 0? How many times is 1 contained in any number? Recite the Table, 


DIVISION. 49 


84, The number to be divided, is called the Dividend ; 
that by which we divide, the Divisor. The result, or 
number obtained by dividing, is called the Quotient. It 
shows how many times the divisor is contained in the 
dividend. 

85. When the divisor is not contained an exact num- 
ber of times in the dividend, what is left over is called 
the Remainder. 5 in 17,3 times and 2 over; 17 is the diy- 
idend, 5 the divisor, 3 the quotient, and 2 the remainder. 


86. The divisor is contained in 
the dividend as many times as it 4 in twelve, ae i: y 
-can be subtracted in succession 8 times. las io EKG 
from the dividend. Dividing is, 
therefore, a short way of performing successive subtrac- 
tions of the divisor from the dividend. 

87. Division is generally denoted by a short horizon- 
tal line between two dots +; the dividend is placed 
before it, and the divisor after it. 12-4 is read, and 
denotes, twelve divided by four. 


Division is also denoted by a line, with the dividend above it and the 
divisor below it; as, 42. When there is a remainder, it is often placed 
over the divisor with this line between, and thus written as part of the 
quotient. Thus: 17 +5 = 32. 

Division is also denoted by a curved line, placed between the dividend 
on the right and the divisor on the left; as, 4)12. 

88. When the divisor is an abstract number, the 
quotient is of the same denomination as the dividend. 
12 men + 4=8 men; 12 apples + 4 = 3 apples. 

89.. AppricaTions oF Drivision.—Division is the con- 
verse of multiplication. The dividend corresponds with 


84. What is the number to be divided called? What is the number we divide by 
ealled? What is the result obtained by dividing called? What does the quotient 
show ?—85. What is meant by the Remainder? Give an example of these definitions. 
—86. How many times is the divisor contained in the dividend? Dividing isa short 
way of doing what ?—87. How is division generally denoted? How is 12+4 read? 
In what other way is division denoted? How is a remainder often written? What 
{s the third way in which division may be denoted ?—88. What denomination is the 
quotient, when the divisor is an abstract number ?—89, Of what is division the con- 
verse? With what does the dividend correspond? ‘The divisor and quotient? 


3 


50. DIVISION. . 


the product, the divisor and quotient with the factors. 
That is, 
dividend = divisor x quotient 382—= 8 x 4 
Hence, dividend + divisor = quotient 382+8 = 4 
dividend -+ quotient = divisor 32-+-4 = 8 

When, then, a product and one factor are given, to 
find the other factor, divide the product by the given 
factor. 

90. We found in § 71 that the whole cost of any num- 
ber of articles equals the cost of one multiplied by the 
number of articles. Hence, 

Divide the whole cost by the number of articles, to 
find the cost of one article. 

Divide the whole cost by the cost of one article, to 
find the number of articles. 


MENTAL EXERCISES. 


How many times is 5 contained in 25? In 40? In80? 8in 
48? 7in 49? 8 in 12? 6 in 54? 7in21? 9in 86? 12 in 
144? 10in 70? 11in110? 12in 48? 9in54? 11 in 121? 

8 in 57, how many times? (Ans. 7 times, and 1 over.) 2 in 
17? 8in14? 4in 89? 5in 83? 6in 45? Yin 18? 8 in 69? 
9in 87? 12im65? 11 in 58? 

Find the quotient and remainder: 87+10. 9+2. 66~+6. 
66-11. 119+12. 100+11. 39+10. 26+12. 62+9, 88+7. 
90+8. 70+6. 47+11. 59+6. 38+12. 

26-3. 57+12. 10+1. 36+10. 1380+12. 24-5. 53+7. 
33+4, 338+8. 72+-9. 101-12. 102-11. 97+9. 19-8. 
40-7. 81-6. 903-12. 140+12. 42+9. 

1. 99 is a product; 11 is one of its factors; what is the other? 
(See § 89.) 

2. How many times is 7 contained in 7x6? 9 in 8x 92 


When a product and one factor are given, how can we find the other factor ? 
—90. When the whole cost and the number of articles are given, how can we find 
the cost of one article? When the whole cost and the cost of one article are given, 
how ean we find the number of articles ? 


SHORT DIVISION. 51 


8. The dividend is 121, the divisor 11; what is the quotient? 

4. If 8 pencils cost 48 cents, how much is that apiece? 
(See § 90.) 

5. How often is 4 times 2 contained in 8 times 5? 

6. How many two-quart pitchers will 24 quarts of water fill? 

7. How many weeks in 56 days, there being 7 days in one 
week? . 
8. How many albums at $3 each can be bought for $30? 
(See § 90.) 

9. If 96 cents are distributed equally among twelve beggars, 
how much will each receive ? : 

10. How many dresses of ten yards each can be cut from a 
piece of calico containing forty yards? | 

11. A merchant lays out $72 for dresses, at $12 apiece; how 
many dresses does he buy? 

12. How many twelve-cent loaves can be bought for 132 cents ? 


91. Suort Diviston.—When the divisor is 12 or less, 
the process is called Short Division. 

Ruix.—Set the divisor at the left of the dividend, with 
a curved line between. Begin to divide at the left. See 
how many times the divisor is contained in each figure 
of the dividend, and write the quotient under the figure 
divided. 

ExampLe.—lIf 4 houses cost $12048, how much do they 
cost apiece ? 


They cost as many dollars apiece as 4 (the number of articles) is con- 
tained times in $12048 (the whole cost). Write the divisor at the left of the 
dividend; begin to divide at the left. 

4 is not contained in 1; see, therefore, how many 
times it is contained in the first two figures. 4 in 12, 4)1 2048 
3 times. Write 3 as the first figure of the quotient, 3012 
under 2, the right-hand figure of the two divided. 

4 in 0,0 times; write it down. 0 must never be 
omitted in the quotient, unless it is the first figure. Ans. $3012. 

4in 4, once. 4 in 8, twice. Ans. $3012. 


91. When is the process called Short Division? Give the rule. Tlustrate the 
vule with the given example. 


UNIVERSITY OF {111 


¢ NQIé 
LIBRARY | a 


“pb? DIVISION. 


EXAMPLES FOR PRAOTICE. 


1. Divide eighteen thousand and six, by six. Ans. 3001. 
2. Divide 100208406 by two. Ans. 50104208. 
3. Divide ninety thousand and sixty-three, by three. 

4. Divide forty-five thousand and five, by five. 

5. Divide 9876543201 by one. 

6. Divide seventy-two billion by 6; by 8; by 9; by 12. 

7. Divide 1800402068 by 2. 


8. Divide twenty thousand eight hundred and four, by 4. 


92. When all the figures of the dividend have been 
divided, if there is a remainder, write it down as such. 
If before this a remainder occurs, prefix it (in the mind) to 
the next figure of the dividend, and continue the division. 

ExaMpLe.—How many stoves, at $12 apiece, can be 
bought with $25009 ? 


As many stoves as $12 (the price of one stove) is contained times in 
$25009 (the whole number of dollars). 

12 is not contained in 2. 12 in 25, twice and 1104 
1 over; write 2 as the first quotient figure under 12) 95009 
the 5, and prefix 1 to the next figure of the divi- ee eres 
dend. 12 in 10, 0 times and.10 over; write 0 An, 2084,1 
in the quotient, and prefix 10 to the next figure, 

0. 12 in 100, 8 times and 4 over; write 8 in the quotient, and prefix 
4 to the next figure, 9. 12 in 49, 4 times and 1 over; write 4 as the 
quotient, and 1 as remainder. Ans. 2084 stoves, and $1 over. . 

This prefixing of the remainder is equivalent to finding how many 
units of the next lower order it contains, and adding thereto the given 
number of units of that order in the dividend. Thus, in the last case 
above, 4 tens (the remainder) = 40 units. 40 units + 9 units = 49 
units,—the result obtained by simply prefixing the 4 to the 9. 


93. Proor or Diviston.—Dividend = divisor X quo- 
tient (§ 89). Hence, Multiply the divisor and quotient 
together » add in the remainder, if there is any. 


Tf this result equals the dividend, the work is right. ea : 

Thus, to prove the last example, multiply the quotient 25008 
2084 by the divisor 12. Addin the remainder 1. The result 1 
equals the dividend ; hence the work is right. 25009 


92. What must be done with remainders that oceur while the division is being 
performed ?—93. How is division proved ? 


LONG DIVISION. 53 


EXAMPLES FOR PRAOTIOE. 


Find the quotient :— 


1. 184766+2. Ans. 92883. 8. 7157842068. Rem. 6. 
9. 83120155. Ans. 62403. 9. 487465834+9. Rem. 4. 
8. 8176483305524, 10. 86782416912. Rem. 5. 
4, 902436421248—6. 11. 269694625~+11. Rem. 2. 
5. 8252'776814796—3. 12. 9038528658+9. Rem. 1. 
6. 864547999637+7. 13. 117850860~+12. Rem. 0. 
7. 1900340128676. 14, 55281904810. Rem. 3. 


94. Lone Drviston.—When the divisor exceeds 12, 
the process is called Long Division. 

95. In Short Division, we subtract and prefix the re 
mainder to the next figure, in the mind. In Long Di. 
vision, we take the same steps, but write down all the 
figures used. 

Exampize.—Divide 361296 by 72. 


In long division, the quotient is set at 


the right of the dividend. Beginning at the : ay 
left of the dividend, take as many ficures as ee SGN ae 
are required to contain the divisor. 72 is ) ( 


not contained in 3, or in 86; itis contained 7x 5= 360 


in 361, 5 times. Write 5 in the quotient 129 
as the first figure. 4 it sx no 
Multiply the divisor by 5; place the _ as cob dab 
product under 361, and subtract. The re- _ 576 
mainder is 1, which (as in short division) Tx8= 576 


we prefix, by bringing down 2, the next 
figure of the dividend. 

Now repeat the same steps. 72 in 12,0 times. Write 0 in the quotient, 
and bring down 9, the next figure of the dividend. 72in 129, once. Write 
1 in the quotient, multiply the divisor by it, and subtract the product 
from 129. The remainder is 57, to which bring down the next figure, 6. 

Repeat again the same steps. 72 in 576, 8 times. Write 8 in the quo- 
tient, multiply the divisor by it, and subtract. There is no remainder, and, 
as all the figures of the dividend have been brought down, the work is 
finished. Ans. 5018. 

361, 12, 129, and 576, are called Partial Dividends. 


94, When is the process called Long Division ?—95, As regards the mode of 
operating, what is the difference between Short and Long Division? Divide 361298 
by 72, explaining the several steps in full, and pointing out the Partial Dividends. 


64 DIVISION. ~ 


96. We may not always, on the first trial, get the 
right quotient figure. 

If, on multiplying the divisor by any quotient figure, 
the product comes greater than the partial dividend, the 
quotient figure is too great, and must be diminished. 

If, on the other hand, on subtracting, we have a re- 
mainder greater than the divisor, the quotient figure is 
too small, and must be increased. 


72) 361296 (6 Thus, in the last example, if we say 72 is contained 
432 6 times in 361, we get a product greater than the 
ice partial dividend, and must therefore diminish the quo- 

tient figure. 

If we say it is contained 4 times, on multiplying 72) 361296 (4 


and subtracting, we get a remainder greater than the 288 
divisor, and must therefore increase the quotient oe ee 
figure. i3 


97. If the divisor is not contained in the partial divi- 
dend, write 0 in the quotient, and bring down the next 
figure of the dividend. If several figures are brought 
down before the divisor is contained in the partial divi- 
dend, write a naught in the quotient for each. 


EXAMPLES FOR PRAOTIOERN. 


Find the quotient. Prove each example (§ 93):— 

. 772326+321. Ans. 2406.| 10. 427854262+-95. Rem. 7. 
. 7050838+-547. Ans. 1289.| 11. 28981539+349. Rem. 4. 
. 7135138+89. Ans. 8017. | 12. 172359694208. Rem. 1. 
. 938986+-74. Ans. 12689.) 13. 9281746+76. 

. 922623+39. Ans. 23657.! 14. 6955070+1682. 

. 961919+106879. Rem. 8.| 15. 3368955+49768. 

. 16360358--6307. Rem. 0.| 16. 52580797048-+8762. 

. 829765804+486. Rem. 8.| 17. 91429832306+45761. 

. 97329468+265. Rem. 3.! 18. 110028314741 +89123. 


96. In the course of the division, what indicates that the quotient figure must be 
fliminished? What shows that the quotient figure must be increased? Give ex- 
amypies.—97. If the divisor is not contained in the partial dividend, what must be 
done? If several figures are brought down before the divisor is contained in the 
partial dividend, what must be done ? 


oOo T eH Be OD DO eH 


LONG DIVISION. ‘BS 


19. 171051980356 -+-300089. 22. 42563008104--82654. 
20. 5763447 +678509. 23. 81000633357+-461305. 
21. 73411659875 + 46398. 94, 24639875555 +5362, 

25. Divide 246515999541 by 28653. Ans. 8603497. 
96. Divide 11963109376 by 109376. Ans. 109376. 
27. Divide 166168212890625 by 12890625. Ans. 12890625. 
28. Divide 1521808704 by 6503456. Ans. 234. 
29. Divide 3278031150 by 46825. Rem. 200. 
30. Divide 4000102955925 by 800095. Rem. 100. 
31. Divide 8976014236 by 12803819. Rem. 978046. . 
32. Divide 243166625648 by 3471082. Rem. 7856. 


83. Divide 9281746 by 27; by 44; by 98; by 294. 
34. Divide 7200651897 by 2498; by 76389; by 82174. 
35. Divide 8976014236 by 298701; by 4853684. 


98. Rule fer Division. 


1. Write the divisor at the left of the dividend. Take 
the least number of figures at the left of the dividend that 
ewtll contain the divisor, and jind how many times tt 78 
contained in them. 

2. If the divisor ts 12 or less, place this first quotient 
Jjigure under the figure divided, or under the right-hand 
jigure of those divided, if more than one are taken. Di- 
vide each figure of the dividend in turn, prefixing the re- 
mainder, tf any, to the next figure of the dividend, and 
writing each quotient figure under the figure divided. 

3. If the divisor exceeds 12, place the first quotient 
Jigure at the right of the dividend. Multiply the divisor 
by it, and subtract the product from the partial dividend. 
Bring down the next figure of the dividend. Find the 
next quotient figure, multiply, and subtract, as before. Go 
on thus, till all the figures of the dividend are brought 
down. 

4, Prove by multiplying quotient and divisor together, 
and adding in the remainder tf there is one. 


56 DIVISION. 


EXAMPLES FOR PRAOTIOE. 


1. The product of two factors is 67048164. One of the factors 
is 9876; what isthe other? (See § 89.) Ans. 6789. 

2. If 1264 acres of land cost $21488, how much is that an 
acre? (See § 90.) 

3. How many barrels of pork, costing $24 a barrel, can be 
‘bought for $95160 2? 

\ 4, If a merchant sells 221988 bushels of corn in 12 months, 
what is the average sale per month? 

5. The earth’s circumference is 25000 miles; how long weuld 
it take to traverse it, at the rate of 200 miles a day ? 

6. The cost of a certain railroad is $8490018. How long is 
the road, if the average cost is $52086 a mile? 

7. A man worth $278195 in real estate, and $49990 in stocks, 
divides the whole equally among his wife, six sons, and four 
daughters. What is the share of each? Ans. $29835. 

8. How many days will 128200 pounds of flour last a garrison 
of 641 men, allowing each man 4 pounds a day % 

9. What number multiplied by 66 will produce 5148? 

10. Divide nineteen million into 9 equal parts. Ans. 2111111}. 

11. If 46 persons consume 158 pounds of flour every day, how 
long will 12482 pounds last them ? Ans. 79 days. 

12. How many firkins holding 56 pounds each will be required 
for putting down 49000 pounds of butter? 

13. There are 5280 feet inamile. How many miles in 971520 
feet? How many in 1948040 feet? 

14. How many bales will 270630 pounds of cotton make, 
allowing 465 pounds to the bale? 

15. If a tax of $44018645 is collected from six thousand and 
forty-five towns, what is the average amount paid by each 
town ? Ans. $7281. 

16. A forest containing 1995 trees was thinned by eutting 
down one tree in seven. How many trees were left ? 

17. There are 6 rows of cannon-balls, each containing 4 piles. 
If there are 76440 balls in all, how many in each pile? 


DIVIDING BY A COMPOSITE NUMBER. aye 


99, Divipine By a CompositE Numser.— When the 
divisor is a composite number, we may either divide by 
the whole at once or by its factors in turn. The result 
will be the same. Division by a composite number may, 
therefore, be proved by dividing by its factors. 

ExAMPLE.—Divide 2232 by 24. 


24=6x4 or, 8 x 3 OV exe @ 


24) 2232 (93 6) 2232 8) 2232 12) 2232 
ie 4) 372 3) 279 2) 186 

72 93 93 93 

13 93 93 93 


EXAMPLES FOR PRACTIOER. < 


In these examples, first divide by the whole divisor; then 
prove the result by dividing by its factors :— 

1. 63 gallons make a hogshead. How many hogsheads are 
there in 9828 gallons? 

2. If 1184 barrels of flour are divided equally among sixteen 
boats, what is the load of each? 

3. If a vessel sails 8168 miles in 82 days, what is her average 
rate per day? 

4, Divide 681660 by 105 (7x35). 
. Divide 160006 by 154 (11x 2x7). 
. Divide 798800 by 843; by 45. 
. Divide 4044425 by 121. Divide 11298 by 42. 
. Divide 2628528 by 56. Divide 83792 by 64. 
. Divide 22500525 by 75. Divide 28416 by 96. 


Cc Ost o> Ot 


100. Tur Truz Remarmper.—In dividing by factors, 
two or more remainders may occur, from which we must 
find the true remainder. Remainders are always units 
of the same kind as the dividends from which they arise. 


99, When the divisor is a composite number, what two modes of proceeding are 
there? How, then, may division by a composite number be proved ?—100, When 
two or more remainders occur, in dividing by factors, how can we find the true re- 
mainder? a eaate this process with the given exaruple, 


58 DIVISION. 


Exampir.—Divide 7464 by 385 (11 x5x 7%). 
Dividing by 11, we get 6 for 


the first remainder. Dividing by 11) "464 Rem. 
11 makes the units in the quo- 5) 678 6 
tient (678) 11 times greater than ) Rohr Snes os 

those of the original number. TY19b) ss SK EL os 


Hence 8, the remainder obtained 


1012 Sone ee 1G 


on dividing this quotient, must 


be multiplied by 11 to make its True rem, 149 
units of the same kind as those 
of the former remainder. In Anan O HA Veen 


like manner, the quotient 1385 is 

made up of units 5 times 11, or 
55, times greater than those of the original number. Hence 2, the re- 
mainder arising from this quotient, must be multiplied by 5 x 11. The 
three remainders being now of the same kind, we add them and get 149 
for the true remainder. Hence, 


To find the true remainder, ine: to the remainder aris- 
ing from the first division, each subsequent remainder 
multiplied by all the divisors preceding the one that pro- 
duced tt. 


EXAMPLES FOR PRACTICE. 


First divide by the whole divisor; then prove the result by 
dividing by its factors, finding the true remainder :— 

1. 223121+27, Fem, 20. 7. 264085--98 (2x 7x77). 
. 258289+-35. Rem, 24. 8. 47484165 (3x11 x5). 
. 83339848, Rem. 38. 9. 89901242 (2x11 x11). 
. 824496 +54, Rem. 10. 10. 91189162 (2x99). 
. 459774 64. Rem. 62. 11. 57212=-198 (8x 6x11). 
. 71515477. Rem. 55. |. 12. 48987-+245 (5x7 x7). 


So Ct He O& bY 


101. NavGuts AT THE RIGHT OF THE Divisor.—When 
there are naughts at the right of the divisor, the opera- 
tion may be shortened. 

Annexing a figure to a number, as we saw in § 25, throws its figures 
one place to the left, and thus multiplies it by 10. Consequently, cutting 
off a figure from the right of a number throws its remaining figures one 


place to the right, and “thus divides it by 10. So, cutting off two figures 
divides by 100; cutting off three, by 1000, &c. Hence, 


101. What is the effect of cutting off a figure from the right of anumber? What 
is the effect of cutting off two figures? Three? 


NAUGHTS AT THE RIGHT, 59 


To divide a number by 10, 100, 1000, cc., cut off as 
many figures at the right of the dividend as there are 
naughts in the divisor. The remaining figures are the 
quotient ; those cut off, the remainder. | 


4200-10 = 420 4200-100 = 42 42001000 = 4, 200 rem. 


102, The principle is the same in the case of any di- 
visor ending with one or more naughts, 
HxaMPLe.—Divide 9710 by 2400. 


Divide by factors. 
2400 = 100 x 24. To Quo. Rem. 
divide by 100, cut offtwo 9710--100 = 97 ........... 10 


figures from the right of 97— 94—= 4...1 x 100 = 100 
the dividend. Dividing 


the quotient thus arising True rem, 110 
by 24, and finding the 
true remainder, we get for Ans, 4, 110 rem. 


our quotient 4 and 110 
remainder, The result is the same as if we had 
9 U0. cut off the two naughts of the divisor and two 
409) 5 i AG (4 9 right-hand figures of the dividend, divided what 
ae remained, and annexed to the remainder the 
110 rem. figures cut off from the dividend for a true re- 
mainder. Hence the following rule :— 


Cut off the naughts at the right of the divisor, and as 
many figures at the right of the dividend. Divide the 
remaining figures of the dividend by those of the divisor. 
ff there is a remainder, annex to it the figures cut off 
from the dividend; if not, these figures are themselves 
ihe remainder. 


349) 10319 (30 1999) 13541 (7 9966) 27994 
102 . 133 3l 
11 2 
Ans, 30, 110 rem. Ans. 7, 247 rem. Ans. 31, 47 rem. 


Give the rule for dividing a number by 10, 100, 1000, &c.—102. Divide 9710 by 
2400, using the factors of the divisor. What ether way is there of arriving at the 
same result? Give the rule for dividing when the divisor ends with one or more 
naughts. 


66 | DIVISION. 


EXAMPLES FOR PRAOTICE: 


Find the quotient :-— 


3. 8875432+~10.  -&, 848670+-560. Rem. 310. 
2. 498268+100. 9. 1993011200. Rem. 101. 
8. 84310006700-+10000. 10. 3315006+850. Rem. 6. 
4. 970000063002-+1000. H. 7294508+900. Rem. 8. 
5. 8186200040--10000. 12. 8400099+280, Rem. 99. 
6. $800800800~+-100000. 13. 1738626+550. Rem. 26. 
7. XDCOGLXXX.-+-X, 14. VOCCC.+ OL. Ans. 36. 


MisceiLanrous Qurstions.—Name the four fundamental operations. 
Ans. Addition, Subtraction, Multiplication, Division ; with these all calcu- 
lations are performed. What is Addition? Subtraction? Multiplication ? 
Division? What operation enables us to find a whole, when its parts are 
given? When the whole and one part are given, what operation enables 
us to find the other part? What is the converse of addition? Of mul- 
tiplication? 

What is the result of addition called? Name the three terms used in 
subtraction. Ans, Subtrahend, minuend, and difference. Define each 
of these terms. Name and define the three terms used in multiplication. 
Name and define the terms used in division. What is meant by the fac- 
tors of aproduct? Which term in division corresponds with the product in 
multiplication? With what do the divisor and quotient correspond? At 
which side do we begin to add? Tosubtract? Tomultiply? To divide? 

What does the sign minus denote? On which side of it must the sub- 
trahend be placed? What does a horizontal line between two dots denote? 
On which side of this sign must the dividend be placed? What does plus 
denote? What does an oblique cross denote? What is the sign of equal- 
ity? How is addition proved? Subtraction? Multiplication? Division? 
In what other way may multiplication be proved? dns. By dividing the 
product by the multiplier; if the quotient equals the multiplicand, the 
work is right. 

What is 2 composite number? Give an example of an abstract com- 
posite number; of a concrete composite number. How may we multiply 
or divide by a composite number? When we divide by factors, how do 
we find the true remainder? What is the shortest way of multiplying by 
10, 100, &c.? How do we divide by 10, 100, &c.? When is division called 
Short, and when Long? What difference is there in the mode ef perform- 
ing the two operations? 


EXAMPLES. 61 


MISOELLANEOUS EXAMPLES. 


1. Find the sum, then the difference, then the product, of 
843 and 8918; divide 8918 by 348. 

2. How many times is 20000 contained in the difference be- 
tween eleven million and eleven billion? Ans. 549450 times, 

3. A United States senator receives $7500 a year. If he 
spends $8 a day, how much of his salary will he save in his six 
years’ term, allowing 865 days to the year? Ans. $2'7480. 

4, If a person hag an income of $3285 a year, how much is 
that a day? 

5. Amile is 6280 feet. How many steps, of two feet each, 
will a boy take in walking 5 miles? Ans, 18200 steps. 

6. Divide the sum of 168483 and 849717 by the difference 
between 97234 and 46324, and multiply the quotient by nine 
times rine. Ans. 1620, 

7. Ifaman earns $1200 a year, and his yearly expenses are 
$860, how many years will it take him to lay up $5440? 

Ans, 16 years. 

8. A farmer buys 75 tons of hay, at $32 a ton. He pays for 
it in wheat, at $2 a bushel. How many bushels of wheat must 
he give? Ans, 1200 bushels. 


What was the whole cost of the hay? How much wheat, at $2 a bushel, will 
pay for it? : 


9. A merchant began business with $86000. At the end of 9 
years he was worth $61875. How much a year had he made? 
10. How many pounds of coffee, at 29 cents a pound, will pay 
for two hogsheads of sugar containing 1160 pounds each, at 19 
cents a pound ? Ans. 1520 pounds. 
11. A person having $2879 in current bills, and $8997 in un- 
current, invests the whole in flour at $9 a barrel; how many 
barrels can he buy? Ans, 764 barrels. 
12. Four partners commencing business put in respectively 
$8650, $9200, $7950, and $3000. At the end of a year the firm 
was worth $37875. Required their gain. Ans. $9075. 
13. Ifa man buys 746 barrels of flour for $8206, what must 


—6©€662 DIVISION. 


he sell the whole for, to gain $1 a barrel? How much is that a 
barrel ? Ans. $12 a barrel. 

14, A person willed $12000 to his wife, $300 to the poor, and 
the rest of his property to his six children in equal shares. If he 
was worth $71870, what was each child’s share? Ans. $9845. 


What was he worth in all? How much of this did he leave to his wife and the 
poor? How much remained? Into how many parts must this be divided? 


15. A lady worth $48530 leaves her servant $550, her brother 
four times that amount, and divides the rest of her property 
equally among her four sons and three daughters. How much 
does each child receive ? Ans. $6540. 


How much does she leave to her servant? To her brother? How much to 
both? How much of her property is left? Among how many is this divided? 


16. Three partners divide equally their yearly profit, amount- 
ing to $17064. One of them divides his share equally among his 
four children; what does each child get? Ans. $1422. 

17. An army of 4525 men had 103075 days’ rations. At the 
end of 21 days, 500 men were captured. How many days after 
that did the rations last ? Ans. 2 days. 


How many rations did 4525 men consume in 21 days? How many rations then 
remained? After the capture, how many men were left? How long would the 
rations left support these men? 


18. A garrison of 842 men had 63472 days’ rations. After 
16 days a reénforcement of 158 men arrived. How long after 
their arrival did the rations last ? Ans. 50 days. 
19. A person bought 97 acres of land at $51 an acre, and 111 
acres at $47 an acre. He paid $9539 cash, and for the balance 
gave 5 horses; what was each horse valued at? Ans. $125. 


What was the.tost of the first piece of land? Ofthe second? Of both? How 
much cash was paid? What remained due? If5 horses were valued at this amount, 
what was each horse valued at? 


20. A hogshead containing 68 gallons of molasses was bought 
for 67 cents a gallon. 7 gallons having leaked out, the rest was 
sold at 76 cents a gallon. What wasthe gain? Ans. 35 cents. 

21. In an orchard containing 659 trees, 41 trees bear no fruit. 
If the income from the orchard is $4944, and the apples bring $4 
a barrel, how many barrels on an average does each bearing tree 
produce ? Ans. 2 barrels. 


MISCELLANEOUS EXAMPLES. 63 


22. A railroad forty miles long cost a million of dollars, all 
but four hundred. What was the cost per mile? Ans. $24990. 
23. The dividend of a sum in division is 4719, the quotient 96, 


the remainder 15. What is the divisor? Ans. 49. 


Subtract the remainder from the dividend, and you have the product of the 
quotient and divisor; then proceed according to § 89. 


24, On dividing 784062 by a certain number, I get 807 for 
the quotient, and 499 remainder. What is the divisor? 

25. If 17 cows are worth $816, and each cow is worth as 
much as 6 sheep, what is the value of one sheep? Ans. $8. 

26. An estate of $25101 was left to a family of four brothers 
and nine sisters. The brothers having given up their share to 
the sisters, how much did each of the latter receive? « 

27. A farmer had 100 hens, four of which died; if the re- 
mainder laid in one week four basketfuls of eggs, consisting of 
120 each, what was the weekly average for each hen? 


Relations of Dividend, Divisor, and Quotient. 


103. The quotient depends on both dividend and di- 
visor. If one of these is fixed, a change in the other 
changes the quotient. But, if both dividend and divisor 
are changed, these changes may neutralize each other, 
and the quotient remain the same. Thus: 


ee eon ea 


Keep the same divisor ; then, 
Doubling dividend doubles quotient: 48 + 6=8 


Halving dividend halves quotient: 12 +6=2 
Keep the same dividend; then, 

Doubling divisor halves quotient: 24—-12=—2 

Halving divisor doubles quotient : 24 +—-3=8 
Doubling or halving both dividend and 48 -12=4 

divisor makes no change in quotient: 12+ 3=4 


103. On what does the quotient depend? If cither dividend or divisor is fixed, 
what is the effect of changing the other? If both dividend and divisor are changed, 
what may follow? With the same divisor, what is the effect of doubling the divi- 
dend? Of halving the dividend? With the same dividend, what is the effect of 
doubling the divisor? Of halving the divisor? Whatis the effect of doubling or 
halving both dividend and divisor ? 


64 DIVISION. 


104, From these examples we conclude that, 

I. With a fixed divisor, multiplying the dividend by 
any number multiplies the quotient by that number, and 
dividing the dividend divides the quotient. 

Il. With a fixed dividend, multiplying the divisor by 
any number divides the quotient by that number, and di- 
viding the divisor multiplies the quotient. 

WI. Multiplying or dividing both dividend and divisor 
by the same number does not change the quotient. 

105, If we multiply one number by another, and then divide the 
product by the multiplier, we have the original number 


unchanged. Multiply 9 by 4; divide the product by 4, 9x4= 36 
and we again have 9. 86+4= 9 


Prime and Composite Numbers. 


168. Every number is either Prime or Composite. 

A Prime Number is one that can not be divided by 
any number but itself or 1, without a remainder; as, 
2,11, 17. | 

A Composite Number is the product of two or more 
factors greater than 1, and is exactly divisible by each 
of its factors. 30 is a composite number = 2x3x535 it 
is, therefore, exactly divisible by 2, 3, and 5. 

107. The first hundred prime numbers are as follows:— 


1/29] 71)113)]173 | 229 | 281 | 849 | 409 | 463 
2/31) 73) 127) 179 | 233 | 283 | 353 | 419 | 467 
3}37| 79}131}181 | 239 | 293 | 359 | 421 | 479 
5|41| 83/137 | 191 | 241 | 307 | 367 | 431 | 487 
7|43|) 89)189/193 | 251 | 811/373 | 433 | 491 
11/47) 97/149 | 197 | 257 | 313 | 379 | 439 | 499 
13 | 53/101 | 151 | 199 | 263 | 317 | 383 | 443 | 503 
17 | 59} 103 | 157 | 211 | 269 | 331 | 389 | 449 | 509 
19/61 | 107 | 163 | 223 | 271 | 837 | 397 | 457 | 521 
2367 | 109 | 167 | 227 | 277 | 347 | 401 | 461 | 528 


104. State the principles deduced from these examples.—105. What is the effect 
of multiplying one number by another, and then dividing the product by the multi- 
plier ?—106. Into what two classes are all numbers divided? Whatisa Prime Num- 
ber? What is a Composite Number ?—107. Mention the first ten prime numbers, 


PRIME AND COMPOSITE NUMBERS. §5 


108. An Even Number is one that can be divided by 
£ without remainder; as, 2, 4, 6, &c. 
An Odd Number is one that can not be divided by 2 
without remainder; as, 1, 3, 5, &c. 
109. A composite number is exactly divisible, 
By 2, when its right-hand figure is 0, or is exactly divisible by 2; as, 
30, 104. 
By 3, when the sum of its figures is exactly divisible by 8; as, 456—the 
sum of its figures (4+5+6 = 15) being exactly divisible by 3. 
By 4, when its two right-hand figures are naughts, or are exactly divisible 
by 4; as, 500, 324. 
By 5, when it ends with 0 or 5; as, 10, 25. 
By 6, when it is an even number and the sum of its figures is exactly 
divisible by 3; as, 744. 
By 8, when its three right-hand figures are naughts, or are exactly divisible 
by 8; as, 17000, 3456. 
By 9, when the sum of its figures is exactly divisible by 9; as, 790146. 
By 10, when it ends with 0; as, 850, 


EXEROISE. 


Tell which of the following numbers are even, and which 
odd; which are prime and which composite. Select those that 
are exactly divisible by 2, by 8, by 4, by 5, by 6, by 8, by 9, 
by 10 :— 

1; 16; 325; 168; 450; 523; 2571; 62875; 9888; 19; 2967; 
85; 29000; 401; 1000101; 8700; 347; 123; 7002; 75408; 6003; 
10101001101201; 655; 10002; 1000. 


Prime Factors. 


110. The Prime Factors of a composite number are 
the prime numbers (other than 1) which multiplied to- 
gether produce it. 2,38, and 11, are the prime factors 
of 66, because 2 x3 x11 = 66. ; 


111. The prime factors of a composite number are 
found by successive divisions. 


108. What is an Even Number? What is an Odd Number ?—1i09. When is a 
composite numbe: exactly divisible by 2? By 3? By 4? By 5? By 6? By 8? By 9? 
By 10?—110. What is meant by the Prime Factors of a composite number? Give 
an example.—111. How are the prime factors of a composite number found ? 


66 DIVISION. | 


Examrrir.—F ind the prime factors of 5460. 


As 5460 is an even number, we divide it by 2. The quo- 
tient, 2730, being an even number, we again divide by 2. 2) 5460 
The next quotient, 1365, is exactly divisible by 3, since the 
sum of its figures is exactly divisible by 3; we therefore 2) 2730 
divide it by 8. The next quotient, 455, is exactly divisible 3) 1365 
by 5, since it ends with 5; we therefore divide it by 5. The 
next quotient, 91, being exactly divisible by 7, we divide it 5) 455 
by 7. The next quotient, 13, is a primenumber. The prime 
factors required are the several divisors and the prime quo- —— 
tient—2, 2, 8, 5, 7, and 18. 13 
Proof.—2x2x3x5x 7x13 = 5460. 


112, Rurz.—1. Zo find the prime factors of a com- 
posite number, divide it by its smallest prime factor ; 
treat the quotient in the same way, and continue thus 
dividing the successive quotients till a prime number ts 
veached. The divisors and the last quotient are the prime 
factors required. 

2. Prove by multiplying the prime factors, and seeing 
whether their product equals the given composite number. 


When a quotient is reached for which a divisor can not readily be 
found, look in the Table on page 64, to sce whether it is prime. If it is, 
the work is done. 


EXAMPLES FOR PRAOTICE. 


. Find the prime factors of 6006. ANS. 2, Oaths Lay Ae 
. Find the prime factors of 16. Of 24. Of36. Of 60. ° 

. Find the prime factors of 72. Of90. Of102. Of111. 

. Find the prime factors of 125. Of 155. Of178. Of 234. 
. Find the prime factors of 309. Of 404. Of 524. 

. Find the prime factors of 1040. Of 1324. Of 6276. 

. Resolve 7498 into its prime factors. Ans. 2, 28, 163. 
. Resolve 28055 into 1ts prime factors. > Ans. 5, 31, 181. 
. Resolve the following numbers into their prime factors: 
- 14641; 78900; 6432: 49750: 390625. 


’ 


Om OT om Ct # CO b He 


pea 
Tt 
(op) 
} 


Find the prime factors of 5460. Prove this example.—112. Recite the rule for 
finding prime factors. When a diyisor can not readily be found, what should be 
done? 


REJECTING EQUAL FACTORS. 67 


Cancellation. 


113. When one set of factors is to be divided by 
another, the operation may often be shortened by first 
rejecting equal factors. 

ExamMpLe.—Divide 6x7x9x5 by 5x3x9x 7. 

We may first multiply the factors of the dividend to- 
gether, then those of the divisor, and then divide the first 
product by the second. 


6x 7x 9x5 = 1890 
Bax Sox Oo xh Sr 
1890 + 945 = 2 Ans. 


But we save work by setting the factors of the dividend 
above those of the divisor with a line between, rejecting 
equal factors from dividend and divisor, and dividing 
what remains above the line by what remains below. 
Thus :— 

6xXx7Tx9x5 
5xX3X9x7 
Rejecting 7, 9, and 5, ee 
ier = ib) bp 


The answer must be the same as before, because rejecting a factor is 
dividing by that factor, and we learned in $104 that dividing both divi- 
dend and divisor by the same number does not change the quotient. 


114. On the same principle, the work may be short- 
ened when the factors of dividend, or divisor, or both, are 
composite numbers. 

ExampLe.—Divide 18 times 21 by 14. 


Arrange as in the last example. Divide 18 9 3 
and 14 by the common factor 2. Then divide 18 x BY 
21 and 7 by the common factor 7%. Multiply; = —_2"" 
ing the factors remaining in the dividend, we 1A 
get 27, Ans. 4 


«27 Ans. 


113. When one set of factors is to be divided by another, how may the operation 
often be shortened? Illustrate this process with the given example.—114. In what 
other case may the work be similarly shortened? Show this with the given example, 


68 DIVISION. 


115. The equal factors thus rejected from dividend and 
divisor are said to be cancelled, and the process is called 


Cancellation. 

Since cancelling is dividing, 1 (nof 0) takes the place of a cancelled 
factor. 

When every factor of the divisor is cancelled, as in the last example, 
the product of the factors remaining in the dividend is the answer. 

For every factor rejected from the dividend we must re- 
ject an equal factor from the divisor, and only one such equal YAN? 
factor. We must not, for instance, cancel two threes in the A\ 418 9 
divisor for one three in the dividend. M | 24 3 

The factors of the dividend, in stead of being placed — 
above those of the divisor, may be set at their right with a Ans, 27 
Vertical line between. Thus :— 


EXAMPLES FOR PRAOTIOER. 


Bring cancellation to bear in the following :— 


1. Divide 2x3x8x5x¥% by 2x4x15. Ans, 14. 
2. Divide 25x 7x11x5 by 55x25 x7, Ans. 1. 
3. Divide 3x 7x2x11x21 by 7x2x38x". Ans. 33. 
4, 40 x 39 is how many times 10 x13? Ans, 12. 
5. Dividend, 1216; divisor, 83 x 22; required the quotient. 
6. How many times is 84x 15 contained in 9x17x38x5x2? 
7. Divide 20 x 36 x 22 x 60 by 8x11 x 100. 


8. Divisor, 5 times 6 times 11; dividend, 6930; what is the 
quotient ? 


9. Divide 99 x 360 x 865 by 11x 738. Ans. 16200. 
10. Divide the product of 17, 10, 16, and 14, by the product 
of 2, 5, 34, 7, and 2. Ans. 8. 


11. How many boxes of raisins containing 12 pounds each, 
worth 20 cents a pound, will pay for 15 boxes of crackers, con- 
taining 16 pounds each, at 18 cents a pound ? Ans. 18 boxes. 

_ 12. How many barrels of coal holding 3 bushels each, at 30 
cents a bushel, must be given for 9 ten-pound boxes of soap, 
worth 12 cents-a pound ? Ans. 12 barrels. 


115. What is said of the equal factors thus rejected? What is this process 
called? What takes the place of a cancelled factor? When every factor of the di- 
visor is cancelled, what will the answer be? How many factors must be cancelled 
in the divisor for each factor rejected from the dividend? In what other way may 
the factors of the dividend and divisor be arranged ? 


4 


GREATEST COMMON DIVISOR. 69 


CHAPTER VIII. 
GREATEST COMMON DIVISOR. 


116. When one number is contained in another with- 
out remainder, the former is called a Divisor or Measure 
of the latter; and the latter, a Multiple of the former. 6 
is contained in 12 without remainder; hence 6 is a divisor 
or measure of 12, and 12 is a multiple of 6. 

117. A Common Divisor, or Common Measure, of two 
or more numbers is any number that will divide each 
without remainder. Their Greatest Common Divisor, or 
Measure, is the greatest number that will divide each 
without remainder. 


2, 4, 6, and 12, are common divisors of 24, 36, and 48. 12 is their 
greatest common divisor. 


118. Numbers that have no common divisor except 1, 
are said to be prime to each other. 


Numbers prime to each other are not necessarily prime numbers. 15 — 
and 28 are prime to each other, yet are not prime numbers. 


119. A divisor of any number is also a divisor of every 
multiple of that number. 3 is a divisor of 6; then it is 
also a divisor of 12, 18, 24, and every other multiple of 6. 

120, A common divisor of two numbers is also a divi- 
sor of their sum and of their difference. 3 is a common 
divisor of 12 and 21; then it is also a divisor of their 
sum (33), and of their difference (9). 

121, To find the greatest common divisor, when the 
numbers are small, resolve them into their prime factors, 
and multiply together those factors that are common. 


116. When is one number called a Divisor or Measure of another? When is one 
number called a Multiple of another? Give examples.—117. What is a Common 
Divisor of two or more numbers? What is the Greatest Common Divisor of two or 
more numbers? Give examples.—118. When are numbers said to be prime to each 
other? Are numbers prime to cach other necessarily prime numbers? Give anex- 
ample.—119. Of what is a divisor of any number also a divisor? Givean example.— 
120. Of what is a common divisor of two numbers also a divisor? Give an example. 
121. How may we find the greatest common divisor, when the numbers are small ? 


70 GREATEST COMMON DEVISOR. 


ExampLe.—Find the greatest common divisor of 72, 
108, and 180. 
12 =O x Deepa 3 
108 = 2x 2 x 3x38 x3 
180 m= 2exXad x 38x 38 Xx 5 


The common factors are 2, 2, 3, and 8; and their product, 36, is be 
greatest common divisor. 


EXAMPLES FOR PRAOTICE. 


Find the greatest common divisor of the following :— 


1. 99 and 72. Ans. 9. 7. 36, 108, and 252. 
2, B4and90. 8. 66, 154, and 220. 
8. 147 and 189. Ans. 21. 9. 120, 185, and 255. 
4, 96 and 264. Ans. 24. 10. 48, 208, and 224. 
5. 120 and 180. 11. 40, 60, 100, and 140. 


6. 144 and 192. Ans. 48. 12. 26, 104, 180, and 234. 


122. When the numbers are large or not easily re. 
solved into factors, we use a different method. 
ExampLte.—What is the greatest common divisor of 


475 and 589 ? 
475) 589 (1 

Divide 589 by 475. If there were no AS 
remainder, 475 would exactly divide both, seat i 
and would be the greatest common divisor. 114) 475 (4 
But, as there is a remainder, divide the last 456 
divisor by it. Again there is a remainder si 
19. Divide the last divisor by it. There is 19) 114 (6 
now no remainder, and 19, the last divisor, 114 
is the greatest common divisor sought. 


That 19 is a common divisor of 475 and 589, is 475 +19 = 25 
proved by dividing those numbers by 19. 589 + 19 = 31 
That 19 is the greatest common divisor is proved thus :— 
Any number that is a divisor of 475 and 589, 
is also a divisor of their difference, or 114 (§ 120), 
also of 4 times 114, or 456 ($119); 
and any number that is a divisor of 475 and 456, 
is also a divisor of their difference, 19. 


ols 


122. When do we use a different method? Illustrate this method with the given 
example. How is it proved that 19 is a common divisor of 475 and 589? How is it 
proved that 19 is their greatest common divisor? 


RULE.—EXAMPLES. 71 


Now, as the divisor of the original numbers must also be a 
divisor of 19, they can have no greater common divisor than 19. 

123. Rutze.—1. Zo find the greatest common divisor 
of two numbers, divide the greater by the less ; if there is 
a remainder, divide the last divisor by it, and so proceed 
till nothing remains. The last divisor is the greatest com- 
mon divisor. 

2. To find the greatest common divisor of more than 
two numbers, proceed as above with the two smallest first, 
then with the divisor thus found and the next largest, and 
soon till all the numbers are taken. The last common 
divisor is the one sought. 


EXAMPLES FOR PRACTIOE. 


Find the greatest common divisor of the following :— 
. 865 and 511. Ans. 73. 9. 1242 and 2328. 
. 864 and 420. Ans. 12. 10. 6409 and 7395. 
. 775 'and 1800. Ans. 25. 11. 10353 and 14877. 
. 2628 and 2484. Ans. 36. 12. 285714 and 999999. 
. 2268 and 8444, Ans. 84. 13. 505, 707, and 4348. 
. 14,18, and 24. Ans. 2. | 14. 154, 28, 848, and 84. 
. 837, 1134, 1847. Ans. 3. 15. 6914, 396, and 5184. 
. 78, 52, 18,416. Ans. 18. 16. 8885, 5550, and 6105. 
Vicia’ far mer wishes to bag 345 bushels of oats, 483 of barley, 
and 609 of corn, using the largest bags of aria size that will 
exactly hold each kind. How many bushels must each bag hold? 
How many bags will he need? Ans. 3 bu. 479 bags. 


The number of bushels each bag must hold, will be the greatest common divisor 
of the given numbers. Then, how many bags holding 3 bushels each will it take to 
hold 345 bushels? How many, to hold 483 bushels? How many, to hold 609 bush- 
els? How many bags will it take in all ? 


18. A man owning four farms, containing 45, 100, 55, and 115 
acres, divides them into equal fields of the largest size that will 
allow each farm to form an exact number of fields. How many 
acres in each field? How many fields does he make? 


NT oP SOD He 


123. Recite the rule for finding the greatest common divisor of two numbers. 
Yow do you find the greatest common divisor of more than two numbers? 


72 LEAST COMMON MULTIPLE. 


CHAPTER IX. 
LEAST COMMON MULTIPLE. 


124, A Multiple of a number is any number that it 
will exactly divide. 4, 6, 8, &c., are multiples of 2. 
Every number has an infinite number of multiples. 

125. A Common Multiple of two or more numbers is 
any number that each will exactly divide. 12, 24, 36, 
&c., are common multiples of 3 and 4. 

126. The Least Common Multiple of two or more num- 
bers is the smallest number that each will exactly divide. 
12 is the least common multiple of 3 and 4. 

127. A common multiple of two or more numbers may 
always be obtained by multiplying them together. If 
the numbers are prime to each other, this product is their 
least common multiple. 

128. A common multiple of several numbers must 
contain all the prime factors of each number taken sepa- 
rately. But a prime factor of one of the numbers may 
also appear in another; and factors thus repeated the 
teast common multiple excludes. Hence, the least com- 
mon multiple is the product of the prime factors common 
to two or more of the numbers, and such factors of each 
as are not common. 

ExamMpLe.—Find the least common multiple of 12, 15, 


18, and 24. 


Write the numbers in a horizontal line. 2 is a prime factor of three 
of them, and will be a factor of the least common multiple; divide by it, 


124. What is a Multiple ofa number? How many multiples has every number? 
—125. What isa Common Multiple of two or more numbers ?—126. What is the 
Least Common Multiple of two or more numbers? Give an example.—127. How 
may 2 common multiple of two or more numbers always be obtained? In what case 
will this product be their Zeast common multiple ?—128, Of what is the least common 
multiple of several numbers the product? Solve the given example, explaining each 
step. 


RULE.—EXAMPLES, %3 


setting down the quotients, and 15, which is not exactly divisible. 2 isa 
prime factor of two of the numbers in the second line; divide by it, setting 
down the quotients, and 15, which is not exactly divisible. 3 is a prime 
factor of all the numbers in the 


third line; divide by it, and set 4 
down the quotients. There ig 2) 12, 15, 18, 24 
no need of dividing further, as 2)6, 15, 9, 12 
no number will exactly divide eS Le ey 
more than one of the numbers 3)3, 15, 9, 6 
in the fourth line. 5, 8, and 2, LOO a eee 


are the remaining factors of the 
original numbers; and the prod- 2% 2X3X5X38X2 = 360 Ans. 


uct of these and the divisors 
(which are the common factors) will be the least common multiple re: 
quired. 2x2x8x5x8x2 = 360 Ans. 


129, When one of the given 
numbers is a factor of another, any 2)42, 15, 18, 24 
multiple of the latter must of course Saag \ Pak LOLA TD. 
contain the former, and the former ___ )15, 9, 12 
number may therefore be cancelled 5, 3, 4 
at the outset. Thus, in the last ex- 


ample, 12, being a factor of 24,may o9yx3x%5x3x4— 360 Ans, 
be cancelled. Proceeding as before, 


we get the same result with less work, 

130, Rute.—1. Write the numbers in a horizontal line. 
Livide by any prime number that will divide two or 
more of them without remainder, placing the quotients 
and the numbers not exactly divisible in a line below. 

2. Proceed with this second line as with the first ; and 
so continue till there are no two numbers that havea com- 
mon divisor greater than 1. The product of the divisors 
and the numbers in the lowest line will be the least com- 
mon multiple, 


EXAMPLES FOR PRACTIOER. 


Find the least common multiple of the following :— 

1, 87 and 41. (See $127.) | 5. 11,77, and 88. _Ans. 616. 
2.23 and 39. Ans. 897.| 6. 24, 180, 45, 60. Ans. 360. 
8. 19, 17, and 5. , 7. 10, 20, 50, 25. Ans. 100. 
4, 2,4,6,and8. (See §129.)| 8. 48, 20, 21,24. Ans. 1680. 


129. When one of the given numbers is a factor of another, how may the opera- 
Kon be shortened?—130 Give tho rule for finding the least common multiple, 


4 e 


74 LEAST COMMON MULTIPLE, 


9. 88, 209, 17, 19, 84. 14. 9, 15, 36, 135, and 162. 
10. 99, 18, 11, 26, and 100, | 15. 144, 48, 80, and 86.. 
11. 34, 88, 75, and 99. 16. 125, 350, 150, and 75. 


12. 875, 10, 8, 12, and 18. | 17. 9, 17, 12, 8, 21, 80, and 16, 
13. 24,20, 18, 16, 15, and 12.| 18, 141, 235, 829. Ans, 4935. 
19. What is the greatest number that will exactly divide 120 
and 150? What is the smallest number they will exactly divide? 
90. Find the smallest number that exactly contains 78, 156, 
and 390. Find the greatest number exactly contained in them. 
91. Find the least common multiple of the first eight even 
numbers. . Ans, 1680. 


CHAPTER X. 
COMMON FRACTIONS, 


131. How Fractions arise.—When a whole is di- 
vided into two equal parts, each of these parts is called 
one half. | 

Half Half 

When a whole is divided -into three equal parts, one 
of these parts is called one third; two are called two 
thirds; &c. 

Third ’ . Third ' Third 


When a whole is divided into four equal parts, one of 
these parts is called one fourth (or quarter); two are 
called two fourths ; three, three fourths; &c. 


Fourth ' Fourth 1 Fourth ' Fourth 


In the same way we get fifths, sixths, sevenths, &c., 
‘by dividing a whole into jive, six, seven, &c., equal parts- 
The name is taken from the number of equal parts into 
which the whole is divided. 


131, How do we get halves? Thirds? Fourths? Fifths? Sixths? Sevenths? 
From what is the name taken ? 


COMMON FRACTIONS. , 75 


132, The value of these equal parts varies according to their number, 
The more parts the whole is divided into, the smaller they must be. One 
half of a thing is greater than one third, one third than one fourth, as is 
shown by the above lines, 


133. These equal parts into which a whole is divided, 
are called Fractions, | 

134, Kinps.—There are two kinds of Fractions, Com- 
mon and Decimal. When we use the word fraction 
alone, we refer to a Common Fraction. 

135, How Common Fractions ARE WRITTEN.—Learn 
how common fractions are expressed in figures :— 


One half 4 | Five thirteenths ys 
One third 4.| Three twenty-seconds shy 
One fourth (quarter) 4 | Twenty sixty-firsts go 


One two-hundredth 4, | Three thousandths IT 
One thousandth sooo | Six twelve-hundredths 3.55 


It will be seen that a common fraction, expressed in 
figures, consists of two numbers, one below the other, 
with a line between, 

The number below the line is called the Denominator. 
It shows into how many equal parts the whole is divided, 
and therefore gives name to these parts. 

The number above the line is called the Numerator. 
It shows how many of the equal parts denoted by the 
Denominator are taken. 

The Numerator and the Denominator, taken together, 
are called the Terms of the fraction. 

is a fraction. 5 and 6 are its Terms. 6 is the Denominator, and 
shows that the whole is divided into stz equal parts, making each part one 


sixth. 5 is the Numerator, and shows that five of these equal parts are 
taken. In reading, name the Numerator first—/ive sizths. 


132. On what does the value of these equal parts-depend? Which is greater, 
one half of a thing or one third? One third or one fourth ?—183. What are the equal 
parts into which a whole is divided called ?—134. How many kinds of fractions are 
there? What are they called ?—135. Show by the given examples how common 
fractions are written. Of what does a common fraction, expressed in figures, con- 
sist? What is the number below the line called? What does it show? What is 
the number above the line called? What does it show? What are the numerator 
and denominator, taken together, called? Give examples of these definitions. 


76 _ COMMON FRACTIONS, 


EXEROISE. 


Read these fractions. Then name the numerator and the de- 
nominator, and tell what each shows :— 
$3 445 3 ss altos robtons WoeouT 


Write the following fractions in figures :— 


1. Ten elevenths. 9. Seventy-three seventy-thirds. 
2. Thirteen halves. 10. One hundred and two four- 
3. Twenty millionths. teen-hundred-and-fifths. 

4, Seventy thousandths, | 11. Sixty-seven forty-thousand- 
5. Eighty sixty-firsts. five-hundredths. 

6. Twelve billionths. 12. Four hundred and two ten- 
7. One hundredth. thousandths. 

8. Four twenty-seconds. | 18. Nineteen six-hundredths. 


136. Drerrnirions.—An Integer is a whole number; 
“ok Ney | 

A Fraction is one or ue of the equal parts into which 
a whole is divided ; as, % 4, 4. 

A Proper Fraction i is one whose numerator is less than 
its denominator; as, 4, 3. 

An Improper Fraction is one whose numerator is equal 
to or greater than its denominator; as, 3, 4. 

A Mixed Number is one that consists of a whole num- 
ber and a fraction; as, 74 (sevenanda half). The whole 
number is called the integral part. 

A Compound Fraction is a fraction of a fraction; as, 
1 of 2, 2 of 8 of 

A Complex Fraction is one that has a fraction in one 
or both of its terms ; as, 

+ One half divided 45 Rm Four and two thirds divided 
9 by nine. 2 by jive sixths. 


136. What is an Integer? What is a Fraction? What is a Proper Fraction? 
What is an Improper Fraction? Whatis 2 Mixed Number? What is meant by 
the integral part of a mixed number? What is a Compound Fraction? What 
is a Complex Fraction ? 


GENERAL PRINCIPLES, 77 


A fraction is said to be dnverted, when its terms are 
interchanged; 4 inverted becomes 4. 

137. Prixciptes.—A fraction indicates division (§ 87). 
The fractional line is the line used in the sign of divi- 
sion +. The numerator is the dividend, the denominator 
is the divisor, the value of the fraction is the quotient. 
Hence the same principles apply as in division (§ 104). 

I, Multiplying the numerator by any number multiplies 
the fraction by that number, and dividing the numerator 
divides the fraction. 

Il. Multiplying the denominator by any number di- 
vides the fraction by that number, and dividing the de- 
nominator multiplies the fraction, 

Ill. Multiplying or dividing both numerator and de- 
nominator by the same number does not change the value 
of the fraction. 

138. A fraction indicates division. Hence, if numera- 
tor and denominator are equal, the value of the fraction 
is 1; because any quantity is contained in itself once. 
If the numerator is greater than the denominator, the 
value of the fraction is greater than 1; if less, less than 1. 


Hence, the value of every improper fraction must be 1 or more than 1; 
that of every proper fraction, less than 1. 


139. Any whole number may be thrown into a frac: 
tional form by giving it 1 for a denominator. 7 = 7. 
9= 4. It is clear that dividing a number by 1 does not 
alter its value. 

EXERCISE. . 

Read the following. Tell what kind of fraction each is. In 
the third line, tell whether the value of each fraction is greater 
or less than 1 :— 


When is a fraction said to be inverted ?—137. What does afraction indicate ? 
What corresponds with the dividend? What, with the divisor? What, with the 
quotient? To what, then, do the principles of divisionapply ? Recite the three prin- 
ciples that apply to fractions.—138. When is the value of a fraction 1? When is it 
greater than 1? When is it less thani? What must be the value of every im- 
proper fraction? Of every proper fraction ?—130. How may any whole number 
be thrown into a fractional form? 


78 COMMON FRACTIONS, 


Zofs,. Fof~ of fy 8,3; 4o0f4%. 12%. 
64 83 6004 #of4 184 ahs 
8 Fr BOe COPIED 14d OF F 
Sy 89 Ff 58 4 FASO. dts. He 
Throw 7 into a fractional form; 19; 871; 1002; 11; 6. 


MENTAL EXEROISES ON FRAOTIONS. 


1. How many halves in 1 whole? How many thirds? How 
many fourths? How many tenths? How many fiftieths? How 
many thousandths? 

2. How many halves in 1? In 2? In 3? In 4? In 10? 
In 100? In 1000? In 100000? How do you find how many 
halves there are in any number? Ans. By multiplying it by 2. 

3. How many thirds in 1? In 2? In 8? In 5? In 12? 
In 100? In 400? In 5000? How do you find how many thirds 
there are in any number? Ans. By multiplying it by 3. 

4, How many fourths in 1? In 2? In 6? In 8? In11? 
In 20? In 200? How many fifthsin1? In9? In4? In800? 
How do you find how many fourths there are in any number? 
How do you find how many fifths? 

5. How many sixths in 1? In 5? In 8? In 10? In 12? 
How many sevenths in 1? In4? In 6? In7? In11? How 
many eighthsin1? In9? In12? In5? In200? Howmany 
ninths in 1?* Ing? In12? 

6. How many elevenths in 1? In11? In 12? How many 
twelfthsin1? In6? In9? In11? In12? Howmany tenths 
inl? In11? In 17? In 176? In 84? In 71? How many 
hundredths in 1? In 5? In 12? In 33? In 45? 

". How do you get half of a thing? Ans. By cutting it into 
two equal parts. How do youfind half of a number? Ans. By 
dividing it by 2. How much is half of 4? Of 6? Of 10? 
Of18? Of24? Of7? (Ans. 34.) Of92 Of112? — 

8. How do you get one third of a thing? Ans. By cutting. it 
into 3 equal parts. How do you find 4}of anumber? Ans. By 
dividing it by 8. Whatisf$of 9? Of 27? Of11? (Ans. 33.) 


MENTAL EXERCISES. 79 


9. How do you get } of athing? How do you find } of a 
number? Hew much is $ of 24? Of 82? Of 386? Of 45? Of 19? 

10. How do you find 4 of a number? Hew do you find ¢? 
4? fot ve? abo? 

1l. How much is } of 402 1 of 42? 4} of 84? 4 of 72? 
Sof 99? +, of 33? 4, of 1382? 3, o0f 110? sy, ef 60? +, of 
96? j,of1322 y,of 1442 1, of 10? ~,of 1602? +4, of 1700? 
rhy of 4500? : 

12. How much is $ of 12? Ans. One fourth of 12 is 8; and 
three fourths are 3 times 3, or 9. 

How muchis# of 6? § of 142 3 0f25? $o0f18? -8 of 44? 
fy Of 24% Yo of40? BR of 482? $of362 #of72? 44 of 1322 

13. What part of 2 is 1? (Ans. 4.) What part of 3 is 1? 
(Ans. $4.) What part of 5is1? What part of 5is 2? (Ans. 2.) 
What part of 5is 32 What part of 7is 1? What part of 7 is 6? 

14, tiew many half-pence in 9 pence? 

15. How many quarters of beef in 12 oxen? 

16. If I cut 10 oranges into sixths, how many pieces have I? 

17. A vessel containing 48 passengers was wrecked. +; of 
the passengers escaped. Tow many escaped, and how many 
perished ? | 

18. If a pound of coffee costs 40 cents, what will half a pound 
cost? 8 ofapound? 8, of a pound? 

19. A boy having 60 marbles lost 35, of them, gave 8 away, 
and kept the rest. How many did he lose, give away, and keep? 

20. If 4 of a ton of coal costs $3, what will a ton cost? Half — 
a. ton? 


140. Fractions may be reduced, added, subtracted, 
multiplied, and divided. 


Reduction of Fractions. 


141. Reducing a fraction is changing its form without 
changing its value. 


146. What operations may be performed on fractions ?—141. What is meant by 
seducing a fraction ? 


80 


COMMON FRACTIONS. 


142, Case 1L—To reduce a fraction to its lowest terms. 
A fraction is in its lowest terms when its numerator 
and denominator have no common divisor greater than 1. 


Exampie.—Reduce 4% to its lowest. terms. 


_Dividing.both numerator and denominator by the same number does 
not alter the value of the fraction (§ 137). We 
therefore divide by their 
common factors in succes- 
sion. Dividing by 5, we 
get-*;. Dividing the terms 


45) 75 
45 


@! 


80) 45 (1 


15 


5 
45 


30 


-15)30(2 
30 


= 2 Ans. 


of this fraction by 3, we 
get #. This is the answer, 
since its terms have no common divisor but 1. 


In stead of dividing as above, we might have 
found the greatest common divisor ($123), and 
This is the best method, 
when the numbers are large. 


divided by it at once. 


Rutz.—Divide numerator and denominator succes- 
sively by every factor common to both. Or, divide them 


at once by their greatest common divisor. 


EXAMPLES FOR PRAOCTIOE. 


. Reduce the following fractions to their lowest terms :— 


1. 


ait Ege 


OD CO oY D> OT BH co bo 


~~ KH 
lem =) 
° ° 


142, 


e 


an 
is 
s 


rahe 
colo 
ols 

° 


—_ 
= ke 


108° 


444 


12. 


$e}. Ans. 

Bu, Ans. 

gi5, =. Ans. 

2 6 

99° Ans. ite 
o 

bits. Ans. P56 


99 1 
oy. Ans. zh. 


CRA coe afer 
e 


23. 


Soe. 
$o8: 

756. 
reer 
3384. 
uae 
36 

000° 
3 78 - 
896 
doe? 


§3 
Seo 


S0n8") 


Ans. 424. 
Ans. p54. 

Ans. $3. 

Ans, $3. 

Ans. §3. 
Ans. 23. 
Ans. $27. 
Ans, $23. 
Ans, 345, 
Ans. 375. 


7 
Ans. Sie 


What is the first case of reduction of fractions? “When is a fraction in its — 
lowest terms ? 


Solve the given example, explaining the steps. 
is shown? Recite the rule for reducing a fraction to its lowest terms, 


What other method 


REDUCTION OF IMPROPER FRACTIONS. 81 


143, Casz II.—TZo reduce an improper fraction toa 
whole or mixed number. 

A fraction indicates division. The numerator is the 
dividend, the denominator is the divisor. To find the 
quotient, that is the value of the fraction, we have only 
to divide, as indicated. 

ExampiLE 1.—Reduce 2 to a whole or mixed number. 

27 = 9 ='3 Ans. 
ExampitE 2.,—Reduce 3,° to a whole or mixed number. 
30 + 9 = 33 = \8hi dash - 
Rute.— Divide the numerator by the denominator. 


If there is a remainder, the answer is a mixed number; if not, a whole 
number. If the answer is a mixed number, the fractional part must be 
reduced to its lowest terms, 


EXAMPLES FOR PRAOTIOE. 


Reduce these fractions to whole or mixed numbers :— 


1’ 1B. 384, Ans. 1044. | 29 5&2. 

2, 53, 16. 200, Ans. 18% | 80, 22642, 
3. 38, 17, 5a. Ans. 452. | 81. 24991, 
4, 144, 18, 2440, Ans. 3928. | 82, 29825, 
5. 194, 19, 2397, Ans. 342,. 33. 19940, 
6. 122. 20. £297, Ans, 9084. 34, 26351, 
7, S24, Q1, 5895, Ans. 924. | 35. 15827, 
8, f7d 29, 2318, Ans, 82.81. | 36. 4faa5, 
9. $9, 23, 20.00, Ans. 8173;. | 87. S400, 
10. 82 24, 142641 Ang, BIGHT. | 88, 29928, 
11. 39 25, 88.00, Ans. 26. | 89. 24926, 
12, 58 26. 14022, Ans. 20031. | 40. 22992. 
13, 134, 07, 11452, Ans, S714. | 41, 427135, 
14, 400, 28, 2482, Ans, BT24, | 42, 48987, 


143. What is the second case of reduction of fractions? What does a fraction in- 
dicate ? With what do the numerator and denominator correspond? How may we 
find the quotient,—that is, the value of the fraction? Give the rule for reducing an 
improper fraction to a whole or mixed number. Give examples, 


4% 


82 COMMON FRACTIONS 


144, Case III.—7Zo0 reduce a mixed number to an tv 
proper fraction. 
_ Exampiye.—Reduce 9% to an improper frac- 
tion. 

The denominator of the fraction being 5, we reduce to “45. fifths, 


or 


fifths. In 1 there are 5 fifths, and in 9 nine times 5 fifths, 3 fifths 
or 45 fifths, 45 fifths and 3 fifths make 48 fifths. dns. 48, — 
Proof. 48 = 48 +5 = 93 48 fifths. 


Ruitze.—1. Multiply the whole number by the denomi- 
nator of the fraction, add in the numerator, and set their 
sum over the denominator. 

2. Prove by reducing the improper fraction obtained 
back to a mixed number. 

145. To reduce a whole number to an improper frac- 
tion with a given denominator, the process is the same, 
except that there is no numerator to addin, Multiply 
the whole number by the given denominator, and set the 
product over the denominator. 

EixaMpLe.—Reduce 9 to fifths. 


Oe ea 4D Ans, 42. 


EXAMPLES FOR PRACTICE. 


Reduce the following to improper fractions; prove each :— 


1. 124. Ans. $4. | B. Ty. 9. 76384. | 18. S4,289.. 
OP 168.” Ans: S804 6. 7 74h. 10. 87647. 14. 484,36. 
3, 24513 (ey exe 11. 81232. 15. 29643. 
4, 19273, Bel. 412, 1284 | 16. Sate 


17. Reduce 18 to a fraction with 7 for its denominator. Ans. 2A. 
18. How many 89ths in 746? In 293? In 450? 

19. Reduce 26 to fortieths. To fiftieths. To sixtieths. 

20. How many quarters of beef in 1225 oxen? 

21. Reduce 887 to nineteenths. To eighty-fifths. 


144, What is the third case of reduction of fractions? Solve and prove the given 
example. Recite the rule for reducing a mixed number to an improper fraction.— 
445. How does the operation differ, when a whole number is to be reduced to an im- 
proper fraction? Recite the tule. Give an example. 


REDUCING TO HIGHER TERMS. 83 


146, Casze IV.—7o reduce a fraction to higher terms. 

A fraction is reduced to lower terms (§ 142) by divi- 
Sion, to higher terms by multiplication. 

ExampLre.—Reduce ? to twenty-fourths. 094.4 — 6 


Multiplying both numerator and denominator by the 
same number does not alter the value of the fraction. 
We therefore multiply both terms by such a number as 
will change fourths to twenty-fourths—that is, 6 (because 
24+4=6) Ans. $8. Ans. x4. 

Rury.—l. Divide the given denominator by the de- 
nominator of the fraction, and multiply both terms by the 
quotient. 

2. Prove by reducing the fraction back to its lowest 
terms. 

Mixed numbers must first be reduced to improper fractions. 


147, A fraction can thus be reduced only to such higher terms as are 
multiples of the original terms. Thus, }? can be reduced to eighths, 
twelfths, sixteenths, &c., but not to fifths or sixths. 


EXAMPLES FOR PRAOTICE. 


1. Reduce -; to seventicths. Ans. 4%. 
2. Reduce the following to 36ths:—%; 443 2; 4. 
0 


8. Reduce to 288ths :— 3%; si; 42; 28; 343 422; 72; 4. 


4, Reduce 144, to twenty-seconds. Ans. 354, 
5. How many 840ths' in 13? Ans. 1818, 


6.. How many 360ths in 17? In 444? In2,%,? In 43? 
7. How many seventy-seconds in 5,2 In2,? In 4%? 
8. Reduce the following to 2460ths :— 47; 12; #54. 


148. Caszr V.—To reduce two or more fractions to 
others having a common (that is, the same) denominator. 

Exampir.—Reduce 3, 4, and 4, to fractions that have 
# common denominator. 


146. What is the fourth case of reduction of fractions ? How is a fraction re- 
duced tolower terms? How, to higher terms? Reduce % to twenty-fourths, explain- 
ing the steps. Recite the rule. What must first be done with mixed numbers ?— 
147. To what higher terms alone can a fraction thus be reduced ?—148. What is the 
fifth case of reduction of fractions? Work out and explain the given example. 


84 


The denominators are 4, 2, and 
Now, a product is the same, in 
whatever order the factors are taken. 
Hence, if we multiply each denomi- 


6. 


COMMON FRACTIONS, 


fond oa 
x XX 


nator by the other two, we shall get 
a common multiple of all three, and this 


Be 2x BS NB 
Bh BX Bo ok 
pt x 8 
2x4X 6 4 
Dia aex es a. 4 
6x* 4x2 4 


| 


| 


| 


6 

8 

4 

3 (As. be changed. 

of : : 

8 its denominator. 


me HA bo 
x XX 


48 
13 (ae 
48). c 


bo om oS 


Hl ll dl 


will be the common denominator, 
But the value of the fractions must not 


We must, therefore, multiply 


each numerator by the same multipliers as 


Henee the ruie :— 


Rvute.—Multiply both terms of each fraction by all 
the denominators except its own. 


Whole numbers must first be reduced to a fractional form, and mixed 
sumbers to improper fractions. 


EXAMPLES FOR PRACTICE. 


Reduce the following to equivalent fractions having a common 
denominator:— 


1. Reduce 
2. Reduce 
3. Reduce 4 
4, Reduce 2 
. Reduce 
. Reduce 
Reduce 
. Reduce 
. Reduce 
Reduce 
Reduce 
Reduce 
Reduce 3 
14. Reduce + 
15. Reduce 
16. Reduce 
17. Reduce 


$ and 3, 

iH and 44. 
% and 32. 

%, 7, and 4. 

$y $y and 4. 

Pai h and $. 

13; Td, and 44. 

4, 14, 12, and 3. 

2%, 4, and 18. 

3, 6, 24, and 88. 

§, ass $y and 24. 

i 4, 4; Fi and $. 

Bo 1%, 4 9 5, and 3 ae 

qs; 13, 43, ae 220. 

34, 8, #, 32, and 100. 

fr; 73, Pal 7, and 3%. 


41, 24, 8, 15, and 23. 


Ans. £9, 
Ans. 439, 434. 


Ans. $48, 343. 
Ans. $8, $8, #4: 


Ans. iu tom Zor: 
4qei vets $38,466. 
Ans. by HES, gh4s, 


Ans. He B, FAA, 34- 
Ans. $455 #55 | 


Common Denom. 32. 
Common Denom. 630. 
Common Denom. 878. 
Common Denom. 900. 
Common Denom. 420. 
Common Denom. 504. 
Common Denom. 840. 

Common Denom. 560. 


What is the rule for reducing two or more fractions to others having a com- 
mon denominator? What must first be done with whole and mixed numbers ? 


LEAST COMMON DENOMINATOR. 85 


149, Case VIL—T7Zo reduce two or more fractions to 
others having the least common denominator. | 

Under the last Case, we found that the common de: 
nominator was a common multiple of the several denomi, 
nators. ‘The least common denominator is the least com- 
mon multiple of the denominators. Find this least com, 
mon multiple, therefore; and then reduce the given frac- 
tions to others that have this least common multiple for 
their denominator, according to § 146, 

EXxAMPLE.—Reduce 3, 4, and 2) 4, 2, 6 
§ to fractions having the least as va 
common denominator. 2x2%3 = 122.0, 


The least common multiple of the J9+-4yx%3—=9 
denominators is 12, which is thereforethe 49.9.4 —6 
least common denominator. To find the : 


several numerators, divide this least com- 12+6x5 = 10 
mon denominator by the denominator of 
each fraction, and multiply the quotient Ans. 755 755) 14- 


by its numerator. 
150. Rutze.—1. For the least common denominator, 
jind the least common multiple of the given denominators. 
2. Hor the new numerators, divide this least common 
denominator by the denominator of each fraction, and 
multiply the quotient by its numerator. 


First reduce the fractions to their lowest terms, and whole or mixed 
numbers to improper fractions. 


EXAMPLES FOR PRAOTICE. 


Reduce the following to equivalent fractions having the least 
common denominator :-— 


3 5 5B 4 5 
a 43 4, and 6° Ans. 5 45. 60? 
12 3 
8 6 5 189 0 
3. 9) 8) and 4 Ans. $33) 529 333° 


149, What is the sixth case of reduction of fractions? Under the last case, what 
did ‘we find the common denominator to be? What, then, will the least common 
denominator be? How, therefore, must we proceed? Solve and explain the given ex- 
ample.—150, Recite the rule for reducing fractions to others having the least com- 
mon denominator, What should first be done with the fractions? With whole or 
mixed numbers? 


86 COMMON FRACTIONS. 


4, +5 q's, and $4. Ans. x5) Peo) Tes 
5. §, #4, and 3%. Ans. 483, 444, tte 
6. 45, Pn and 6%. Ans. $f5, $48) 7s 50s 
7. 4, $, 34 and}. Ans. 29, 49, 195, 42, 
8. a5, 3, 4, and s4. Ans. £25; vas, Fos HLo- 
9. & 4, $y and A ts Ans. 3h es $2, ‘ < 
10. 3%, qo, £0, and 50. Ans. $§, $$) vo) 130% 
11. 3s, a5, 7s, and 13. Least Com. Den. 144, 
12. 2, 8; $, &) soe Least Com. Den. 60. 


13. 3 g, 4; g, 10) and Tio 

14. 4, $, 4, 4,4, + 4, and 4. 
15. ¥, 4, to Ts %, 1%, and $e 
16. 24, 17%, 5, 373, and 43. 


Ad@ition of Fractions, 


151. Like parts, such as halves and haives, thirds and 
thirds, can be added, just as we can add pears and pears, 
dollars and dollars. Unlike parts, such as halves and 
thirds, can not be thus directly added, any more than we 
can add pears and dollars. 

ExAMPieE 1.—Add 5 sixths and 3 sixths. Ans. 8 sixths. 


The denominators being the same, we add the numera- sg Paes 
tors, and place their sum over the common denominator. 6p 5 aes8 


-Examepre 2.—Add 5 sixths and 3 fourths. 


The denominators being different, we can not 


a 19 add the numerators, and call the sum 8 sixths or 8 
: ee fourths. But, if we reduce the fractions to others 
/ erm a having a common denominator, we can then add, as 


ver chs ERE in Ex. 1. 12 being the least common denominator, 
wy 18 A reduce the given fractions to twelfths. 


ExamrLe 3.—Add together. 4, 24, 4%, and 1. 


Add the fractions, as in Ex. 2: qtt4t¢2=15 
Add the whole numbers: 2+44+1= 7 
Add these two sums: Ans. 85'5 


151. Can we add like parts, snch as halves and halves, directly? Can we add 
nnlike parts, such as halves and thirds, directly? Add Sand 3, Add & and}. Solve 
Example 3, , 


ADDITION OF FRACTIONS. 87 


152, Rurz.—1. When the fractions have a common 
denominator, place over it the sum of their numerators. 
When not, after reducing them to their lowest terms, 
change them to equivalent fractions having the least com- 
mon denominator, and add as above. Reduce the result 
to tts lowest terms, or to a whole or mixed number, as may 
be necessary. 

2. To add mixed numbers, find the sum of the fractions 
and whole numbers separately, and add the results. 


EXAMPLES FOR PRAOTIOR. 


Find the sum of the following fractions :— 


Vee tg ba Ans. 14. T $4+34+44. Ans. 24. 

Chap ene ty 8, $4244. 

3. got kot 3b 9. $+3+$+70- 

4, pe-+hh4-244+129, 10. 2464442. 

5. 243499447, 11. $5+494+28+2. 

6. F+24+84+4+844. 12, 3-+3+42+5%. 

13. Add together 5, 32, 42, and 2. Ans, 2397, 
14, Add together 2, 44, +, and #?. Ans. 4,44. 
1b. Find the sum of ;3,, 104, 122, and 2H. Ans. 26,3;. 
16. Find the sum of 4,5, 474, 4,4, and +. Ans. 12493, 
17. What is the value of 3$+6449441}4? Ans, 1933. 
18. What is the value of 334+%+12+%+41}? Ans. Tee. 


19. Add =; and 74. Add #%and4. Add 8 and 3. 

20. What is the cost of four fields, containing respectively 
43, 24, 32, and 142 acres, at $25 an acre? Ans. $300. 

21. Bought $104 worth of cloth, $52 worth of lace, $18.23, 
worth of velvet, and $94 worth of muslin. How much change 
must I receive for a $50 bill? Ans. $7. 

22. How many times can four baskets, holding respectively 
375, 24, 14%, and 2% pecks, be filled from a pile containing 20 
pecks of potatoes ? Ans. Twice. 


152. Recite the rule for the addition of fractions. Recite the rule for the addition 
ef mixed numbers, 


88 COMMON FRACTIONS. 


Subiraction of Fractions. 


153. Case I.—Zo subtract a fraction from a fraction. 
As in addition, so in subtraction, if the fractions have 
not a common denominator, they must be reduced to 
equivalent fractions that have. 
ExAMPLe 1.—From 5 sixths take 4 
aixths. §—4 = 4 Ans. 
4 sixths from 5 sixths leave 1 sixth. Ans. 3. 
Exameie 2.—From 5 sixths take 3 fourths. § = 4$ 
We can not directly take fourths from sixths; but (12 #2 
being the least common multiple of the denominators) we _ 
can reduce both to twelfths, and then subtract. Ans, abs 
154. Rute.— When the fractions have a common de- 
nominator, place over tt the difference of their numerators. 
When not, reduce them to equivalent fractions having 
the least common denominator, and proceed as above. 


EXAMPLES FOR PRAOTIOER. 


Find the value of the following :— 


1. §— 3. Ans. 4.| 6. 4—4. Ans. #;. | 11. €— %. 
at ete Yea ang 197119" § 
3. qt im aise 8. 3 — 35° Ans. Bo: 13. +5 = rea 
4, 42— 4. 9. 2 — dh. Ans. #. | 14. 44 — #5. 
5. Pr pee fia 10 15 a is. Ans, 4. 1D, 2s —_, qs 


155. Case IL—7o subtract a fraction from a whole 
number. 


ExampLe.—From 3 take %. 3 = 22 
, Take 1 of the 3 units, and reduce it to ninths. From 3 
the § thus obtained subtract %, and bring down the 2 units, aoe 
Ans. 25. Ans. 24 


153. In subtraction of fractions, what is it necessary to do if the fractions have 
not a common denominator? From § take ¢. From § take 3.—154. Kecite the rule 
for subtraction of fractions,—155, From 8 take 3. 


SUBTRACTION OF FRACTIONS. 89 


Ruitu.—Reduce 1 to a fraction having the same de- 
nominator as the given fraction. Hrom this subtract the 
given fraction, and annex the remainder to the given 
whole number less 1. 


EXAMPLES FOR PRAOTICE. 


1. 2—4. Ans.1}.| 5. 1—4. Ans. 4.} 9. 2—15. 

2, 14— qr: 6. 8— Sie Ans. Tiss. 10. 1— —t oF. 
10) | 76-83, Ans, 4§8.] 11. 19—843,. 

a 8. 11—3%. Ans. 1042. | 12. 28—JA48.. 


156. Casn IIL—7o subtract one mixed number from 
another. 


ExaMPLe 1.—From 43 take 14. Po t= vs 
3, the fraction of the subtrahend, being less 19 — 3, — a5 
than %, the fraction of the minuend, we subtract A 1 = 3 
fraction from fraction, and whole number from 
whole number, and combine the results. Ans, 34% 


Examprtr 2.—From 41 subtract 14. 


Reducing the given fractions to hia hay- 
ing a common denominator, we get 75 and +3 


41 = 43, — The numerator of the fraction in the gubtve 

341 Pee Qik hend being the greater, we can not proceed as 
Finer ees in the last ; Example. 

rig = 119 From 4, the whole number of the minuend, 


we take 1, and, reducing it to fifteenths, add 
the result to the 5%; of the minuend. From 1%, 
thus obtained, subtracting 49%, the fraction of 
the subtrahend, we have 3%; for the remainder. 
Ans, x. Then, proceeding to the whole numbers, 1 
from 3 leaves 2. Combining the results, we 

have 258. Ans. 


157. Rutu.—Reduce the fractions, if necessary, to 
others having a common denominator. If the numer 
ator of the fraction in the minuend is equal to, or greater 
than, that in the subtrahend, subtract fraction from 
Sraction, and whole number from whole number. If not, 
take 1 from the whole number of the minuend, and reduce 
it with the fraction of the minuend to an improper frac- 
tion. Then subtract as above, 


Recite the rule for subtracting a fraction from a whole number.—156, From 43 
subtract 13. From 4; take 13.—157. Recite the rule for subtracting one mixed num- 
ber from anether. 


90 COMMON FRACTIONS. 


EXAMPLES FOR PRAOTICE 


1. 66 —23., Ans. 4,.| 5. 84 —22. Ans. B44. 
2. 24—13. Ans. 44. | 6. 54 —54. Ans. a5. 
3. 43% — gh. Ans. 312.17. 8348—214. Ans. 443. 
4, 20044 — 98,3, 8. 4732 — 31,4. Ans. 1673. 
9. From 54+ 93 take 48. Ans. 1075. 

10. Take +44 from $+4, : Ans. 3842. 
11. Take 4443, from 64475. Ans. 5443. 
12, From $+1$+£+41} take $4+4+4+44+4. Ans. 34. 
13. From 5+ take 51+ 34. Ans. 335. 
‘14, Take the sum of 14 and 2} from 24+ 8}. Ans. 24. 
15. From 181-20 subtract +. Ans. 8;3;. 
16. From 44080 subtract 494. Ans, 428. 


Multiplication of Fractions, 


158. Case I—Zo multiply a fraction by a whole 
number. 

We found in § 137, that multiplying the numerator or 
dividing the denominator by any number multiplies the 
Sraction by that number. Hence the rule :— 

Ruiz.— Divide the denominator of the fraction by the 
whole number, when it can be done without a remainder ; 
when not, niultiply tts numerator, 


ExamMp.e 1.—Multiply 3 by 5. 
25 is exactly divisible by 5. Divide it. Ans. %, 


EXAMPLE 2.—Multiply »%& by 6. 
25 is not exactly divisible by 6. Multiply the numerator. Ans. }§. 


It is best to divide the denominator when it can be done, because the 
answer is thus found in its lowest terms, 

Dividing the denominator increases the size of the parts as many times 
as there are units in the divisor. Multiplying the numerator increases the 
' number of parts as many times as there are units in the multiplier. 


158, Recite the rule for multiplying a fraction by a whole number. Solve the 
examples given. Why is it best to divide the denominator when it can be done? 
What is the effect of dividing the denominator? What is the effect of multiplying 
the numerator? 


7 


MULTIPLICATION OF FRACTIONS. 91 


159, Multiplying a fraction by its own denominator 
gives the numerator. Thus: 7 x 9 = 7 = 7 Ans 


EXAMPLES FOR PRAOTIOE. 


Find the value of the following :— 


1. 44x24, Ans. 17. 6. 455 x8. ll. A, x 12. 
2.%4x3, Ans. 3h 7. Pex 14. 12. 34,x18. 
8. 4x5. Ans. 22. 8. gy X13. 18, 535x125. 
4, 8x14, Ans. Bh. 9. 4x10. 14, $88 x 288, 
5. 48x49. Ans. 64. | 10. 4x4. 15. x15. 


160. Casz II._— 70 multiply a mixed by awhole number. 


Rurz.— Multiply the fractional and the integral part 
separately, and add the products. 


ExampLe,—Multiply 34 by 7. . 
Multiply the fractional part; Rx 7 = 35 = BR 
Multiply the integral part: at: = 21 
Add the products : 26§ Ans. 


EXAMPLES FOR PRAOTIOR. 


1. Multiply 44 by 8. By 4. By 5. By 7. By 14. 

2. What cost 8 dolls, at $151, each ? Ans, $84. 
8. Multiply 234-34, by 4. Ans, 22%. 
4, At $68 apiece, what cost five coats? Ans. $334. 
5. Multiply 63, — 2,9 by 7. Ans, 224, 
6. Multiply 12 times 4% by 10. Ans. 585, 


7. How much cloth in 4 pieces, each containing 89% yards? 


161. Casr IIL.—7To multiply a whole number by a 
fraction. 

Multiplying by 4 is taking 4 (or dividing by 2) ; mui- 
tiplying by 4 is taking 4 (or dividing by 8); and gene- 


Give the rule for multiplying a mixed number bya whole number. Multiply 83 by 7. 
—161, What is meant by multiplying by one half? By one third? 


92 COMMON FRACTIONS, 


rally, multiplying by a fraction is taking such a part as 
is denoted by the fraction. 


ExampLe.—Multiply 19 by %. 3) 19 
Multiplying 19 by 2 is taking 3 of 19. One third of 64 
19 is 63, and two thirds are twice 64, or 123. Ans, 123, 2 


Here we have divided the whole number A 
19 by the denominator 8, and then multiplied 475. 125 
9 by the numerator 2; but the result is the 
same if we multiply first and then divide, and it often saves 
3) 38 trouble to do so. Hence the rule :— 

Ans, 12% Rutz.— Multiply the whole number by the 

numerator of the fraction, and divide by its 
denominator. 


First see that the fraction is in its lowest terms. 


EXAMPLES FOR PRAOTICE. 


Find the value of the following :— 
1. 47x 8, Ans, 414, 4, 221 x 4%. 7. 49x 20. 


2. 93x 4. Ans. TT. 5, 458 x 43. 8. 2846 x44, 
3. 69x %. Ans. 161. 6. 598 x 4h. 9. 6789 x HH. 
10. Multiply four billion by $29. Ans. 440044004364, 


11. Find the product of 19 million and $88. Ans. 168888888. 

12. A century is 100 years) How many years in # of 10 
centuries? In 4 of 20 centuries? 

18. How many feet in % of a mile, there being 5280 feet in a 

mile? How many feet in ;% of a mile? 

14. A merchant owes $20000. How much is his property 
worth, if it amounts to # of his debts? Ans. $85712. 

15. The moon is 240000 miles from the earth. If it were but 
zx of that distance, how far from the earth would it be? 


162, Casz IV.— To multiply a whole by amixed number. 


Rute.— Multiply by the fractional part and the whole 
part separately, and add the products. 


In general, what is multiplying by a fraction? Multiply 19 by 3, in both the 
ways shown above. Recite the rule for multiplying a whole number by a fraction. 
—162. Recite the rule for multiplying a whole number by a mixed number. 


MULTIPLICATION OF FRACTIONS. 93 © 


Examrite.—Multiply 458 by 93. es" 
Multiply 458 by 3 ($161): 3434 
Multiply 458 by 9: 4122 
Add the products : 44654 Ans. 


EXAMPLES FOR PRAOTIOE, 


1.19x4,%. Ans. 8243. | 4. 875x634. | 7. 84x. 

2. 62x124. <Ans. 752%. 5. 741 x 839. 8. 9080 x 54. 

8. 86x74. Ans. 6284. | 6. 219 x 9.8. 9. 79x 3738. 

10. Multiply 45 thousand by 81,%,. Ans. 8652941,3,. 

11. How many feet in 320 rods, there being 164 feet in one rod? 

12. At $75 an acre, what is the cost of three lots containing 
respectively 34, 44, and 54 acres? Ans. $9583. 


1638. Case V.—7o multiply a fraction by a fraction, 
or to reduce a compound fraction to a simple one. 

Multiplying by a fraction, we learned in § 161, is 
equivalent to taking such a part as is denoted by the 
fraction. Multiplying %, 4, and } together, is equivalent 
to taking 2 of 4 of 7. The same process is therefore used 
in multiplying fractions together and in reducing com- 
pound fractions to simple ones. 


EXAMPLE 1.—Multiply 2, 4, and 4 together. 

These fractions indicate division. The numerators are the dividends ; 
the denominators, the divisors. Multiply the numerators together to find 
the total dividend, and the denomina- 
tors to find the total divisor. Then 
set the former product over the latter gx GX $= 235 Ans. 
in the form of a fraction. 


164, As in division ($113), cancelling often shortens 
the operation. By first cancelling the equal factors com- 
mon to any numerator and denominator, we get the 
answer at once in its lowest terms. 

Solve the given example.—163, To what is multiplying by a fraction equivalent ? 
In what two operations, therefore, is the same process used? Explain Example 1.— 


164. How may the operation often be shortened? Whatdo we gain by first cancelling 
equal factors? 


04 COMMON FRACTIONS, 


Ex. 2.—Reduce § of ,3, of } of +4 to a simple fraction. 


Cancel 5 and 5. Bea 8 in the second numerator and first denomi- 
nator. Cancel the 2 then re- 
maining in the first denomi- 


2 
nator, “and 2 in the third 
numerator, Cancel 7 in the p a 3 OF 2 ae 1A oe 2 “on 
fourth numerator and second 6 Ag 9 5 63 : 
denominator. Then multiply yD q 


the remaining factors, as in 
the last Example. 


Ex. 3.—Multiply together 24, 122, 4, and 11. 


Reduce the mixed numbers to improper fractions. Throw the whole 
number into a fractional form, by giving ‘it 1 for its denominator. Then 
proceed as in Example 2, 


i 
PX gS P= 10h ae 
2 2 

Cancel 17 in the first numerator and second denominator. Cancel 4 
in the third numerator and first denominator. Cancel 9 in the second 
numerator and third denominator. Multiply the remaining factors. Re- 
duce the improper fraction obtained to a mixed number. 

165. Rute.—1. Cancel factors common to any nume- 
rator and denominator. Then multiply the numeratore 
together for a new numerator, and the denominators Sor 
a new denominator. 

2. Whole numbers must first be reduced to a fractiona: 
form, and mixed numbers to improper fractions. Reduce 
the result, when necessary, to a whole or mixed number. 


EXAMPLES FOR PRAOTIOER. 


Find the value of the following :— 


1. 38x 38. Ans. $. 5. }xExé. Ans, $. 
2. 48x 28. Ans. 43. 6. £x4x-f. Ans. $9. 
8. 8, x 44. Ans. +. T. SX gt MAO eee 
4, 4x4xH. 8. +x 92x40. 

9. Reduce to a simple fraction 2 of $ of % of 4. Ans. 35. 


2 'Go through Example 2. Explain Example 3.—165. Recite the rule for multiply- 
ing a fraction by a fraction, or reducing a compound fraction to a simple one. 


MULTIPLICATION OF FRACTIONS. 95 


10. Reduce 4 of § of 4° of 4% to its simplest form. Ans. 138. 


11. Find the product of # of 4$ and 3 of 12. Ans. +45, 
12. Multiply 2 of 3 of 3 by 4 of 44. Ans. $23;. 
18. Multiply 4 of 44 by 314. \Ans. 24, 


14, Multiply 74 by 34. 144 by 5. Add the products. 
15. Multiply 64 by 24). 4 by 8?. Add the products. . 
16. Find the difference between 3} x 8; and 55x 2,4,. Ans. 11. 


17. Find the value of $x 9, x 4% x x7. Ans. 7. 
18. Reduce # of } of 3 of +4 of 14 of § of §. Ans. x. 
19. How much more is 6 times $ than 18 times 3? Ans. 24. 
20. Multiply 8$+383+4387% by 14444. Ans. 208. 
21. Reduce $8 of $2 of 42 of 44 of 124. : Ans. 43. 
22. Reduce 1§ of +4 of $ of 44 of 34. Same ans. 
23. Multiply 3x38x x} by 7. Sor both. 


Division of Fractions, 
Reduction of Complex Fractions. 


166. A fraction divided by a ; 
fraction may be expressed in two = Three fourths Ase t 
ways: with the sign of division, or oe eae: 
in the form of a complex fraction. ae 
Whichever way the division is ex- 
pressed, the operation is the same. Hence, to reduce a 
complex fraction to a simple one, take the denominator 
as a divisor, and proceed as in division of fractions. 


167. Casz l.—To divide a fraction by a whole number. 

We found in §137 that dividing the numerator or 
multiplying the denominator by any number divides the 
Sraction by that number. Hence the rule :— 

Rurz.— Divide the numerator of the fraction by the 
whole number when té can be done without a remainder ; 
when not, multiply its denominator. : 

166. In what two ways may a fraction divided by a fraction be expressed ?—167. 


What is the first case of division of fractions? Recite the rule fer dividing a fraction 
by a whole number. 


96 COMMON FRACTIONS. 


Dividing the numerator diminishes the nwmber of parts ag many times 
as there are units in the divisor. Multiplying the denominator diminishes 
the s2ze of the parts as many times as there are units in the multiplier. 


Exampie 1.—Divide 35 by 6. 

36 is exactly divisible by 6. Divide it. af — 6 = § Ans... 
36 

ExaMPLe 2.—Reduce ag to a simple fraction. 


36 is not exactly divisible by 5. Multiply 36 
the denominator. a As 


EXAMPLES FOR PRAOTIOER. 


Find the value of the following :— 


* 


pes aiihrot 9 1, EES, 13. Reduce et Ans. Ay. 
2. s4%+8. 8, 2988525, 36 
Bes 5940-2 

B. gig t-5. 9, ABsAP 48. | Reduce 2H Ae 44. 

4, JA+T. 10, 22$62-+-67, 

5. AGF8+12. | 1). 46444--91. 42 te, $0 
15. Reduce —- ; IO) 

G. 4949010, | 12, 24388...86, Sa 6 ea 


168. Casz Il.—7o divide a mized by a whole number. 
Ex. 1.—Divide 819% by 9. 


Divide the integral part: 819 + 9 = 91 
Divide the fractional part: - $+9 = 3% 
~ Combine the quotients: 91.2, Ans. 
8467 : : 
Ex, 2,—Reduce sie to a simple fraction. 
The numerator of the complex fraction 8) 84673 
is the dividend, the denominator the divi- 10584 
sor. Divide 84678, the integral part of 10584, 1 rem, 


the dividend, by 8. 1 remains, which pre- 52 eee 

fixed to the fraction makes 12, of De I1G=4¢ $78=75 

viding 4 by 8, we have 4. Combining 

the quotients, we get 10584.2, Ans, Ans. 10584¢5 
Ruiz.—1. Divide the integral and the fractional part 


separately, and combine the quotients. 


What is the effect of dividing the numerator? Of multiplying the denominator ? - 
Solve the examples.—168. What is the second case of division of fractions? Explain 
the given examples. Recite the rule for dividing a mixed by a whole number. 


DIVISION OF FRACTIONS. 


97 


2. Lf, on dividing the integral part, there is a renvain- 
der, prefix it to the fractional part, reduce to an improper 
Fraction, divide as in Case I., and combine this quotient 
with that obtained by dividing the integral part. 


EXAMPLES FOR PRAOTICE. 


1. Divide 84 by 20. Ans. #. 9, 59403,+-12. 
2. Divide 3,2, by 8. Ans. #4. | 10. 89913425. 
8. Divide 68 by 9. Ans. #8. | 11. 95094352. 
4, Divide 94, by 3. Ans. 34; | 12. 10013+-10. 
5. Divide 8345 by 2. Ans. 4,4. | 18. 9107464. 
6. Divide 93 by 7. Ans. 144. | 14. 784144~38. 
ve 


Reduce 878 


Ans. 1743.-| 15. Reduce 


faa 
44 


a Ans. 144. 


Ans. 495 Ay. 

Ans. 35944, 
Ans. 18245. 
Ans. 100z%. 
Ans. 142-8;. 
Ans, 20644. 


8. Reduce Lee Ans. 247%. | 16. Reduce eas: Ans. Ist. 


17. Reduce 4, 25§, 107, 14444 28879 45158, 26714 
Gi 40," BG, i Odie 986 Aran 88 


169. Casz IIL.—T7o divide a fraction, whole, or mixed 


number, by a fraction or mixed number. 


Ex. 1.—How many times is ? contained in 3? 


+ is contained in 1, 7 times, In 2 it is contained 3 
of 7 times, or #1 times. 

But # is twice as great as +, and hence is contained 
only half as many times. 4 of 4+ = 24 = 2,4; Ans. 


Now, what have we done to the dividend 3, to pro-_ 


duce the quotient #4? We have multiplied it by the 
divisor inverted. Hence the rule :— 


3 — 

gxt = ah 
ett Sie 
Fxg = Hh 
3 — 214 
$xg=it 


Rurz.—1. Multiply the dividend by the divisor in- 


verted, 


2. Whole and mixed numbers must first be reduced to 


improper fractions. 


169. What is the third case of division of fractions? How many times is ? con- 
tained in 3? What have we done to the dividend, to produce the quotient? Reette 


the rule for dividing one fraction by another 


5 


98 ~ COMMON FRACTIONS. 


EXxaMPLE 2,—Reduce mt to its simplest form. 


Reduce the numerator to an improper fraction ; 31 


ze 


i 
Multiply the dividend by the di- dt28 a ee : 

visor inverted, cancelling common = tes 1S ae 

factors. Reduce the result to a 4 ner 

mixed number. 2 


Tl 


Reduce the denominator to an improper fraction: 2+4 


ExaMPLeE 3.—Divide 4 by 2. 


The denominators, being the same, are can- 
celled when the divisor is inverted, and we have 4 
only to divide 4, the numerator of the dividend, 
by 2, the numerator of the divisor. Hence, 4 
When the Fractions have a common denominator, — 
reject it, and divide the numerator of the dvi- 1 
dend by that of the divisor. 3 


EXAMPLES FOR PRACTICE. 


Find the value of the following :— 

1. +4. Ans. 3. 4. Divide +4; by 4 7. a+ Z 
2. +4. Ans, 24. 5. Divide $$ by 23. 8, 48+ 28. 
3. +44. Ans. 14. 6. Divide 8&8 by #4. Se 


Reduce the following to their simplest forms :— 


1 of 
10. Reduce = Ans. $. | 14. Reduce = is us Ans. #5. 


65 
11. Reduce aa Ans. 144. | 15. Reduce - i . Ans, 108. 
4 4 of 2 
$44 
12. Reduce Sf. Ans. 80%. | 16. Reduce + BO Ans. 4. 
ny © 
612 bane E 
13. Reduce %. Ans. }. | 17. Reduce 224. Ans. 4. 


18. How many times can a pitcher holding 14 quarts be filled 
from a pail containing 5 quarts? 


Solve and explain the given examples. When may cancellation be brought 


to bear? When the fractions have a common denominator, what is the best mode of 
proceeding ? 


MISCELLANEOUS QUESTIONS. 99 


' 19. What is the rate per hour of a boat that goes 23043 miles 

in 18? hours? Ans. 12,5; miles. 

20. If 837% yards of calico are used in cutting three dresses of 
equal size, how many yards are there in each dress? 

21. Five and a half yards make a rod. How many rods in 
838 yards? 

22. If a man makes $144 on every table he sells, how many 
tables must he sell to make $273? 


MisceLLaANnEous -Qurstions oN Fractrons.—What does the word /ra¢, 
tion come from? Ans. From the Latin word fractus, broken, because a 
fraction indicates the breaking up or dividing of a unit into equal parts. 
What is meant by the terms of a fraction? With what do they correspond 
in division? What is the difference between a proper and an improper 
fraction? Which is the greater? Which is greater, a proper fraction or 
a mixed number? Which is greater, | or}? When we increase the de- 
nominator of a fraction, do we increase or diminish its value? Which is 
greater, } or 3? When we increase the numerator of a fraction, do we 
increase or diminish its value? Which is greater, 4 or}? What kind 
of fractions are these ? 

What is meant by reducing a fraction? Mention all the cases of re- 
duction of fractions that you can remember. How do you reduce a frac- 
tion to its lowest terms ? How do you reduce an improper fraction to a 
whole or mixed number? How do you reduce a mixed number to an im- 
proper fraction? How do you reduce a compound fraction to its simplest 
form? How do you reduce a complex fraction to its simplest form ? 

When we take } of a number, do we multiply or divide by }? What 
is dividing by } equivalent to? Dividing by 3 is equivalent to multiply- 
ing by what? Multiplying by 3 is equivalent to dividing by what? Does 
multiplying a number by a proper fraction increase or diminish it? 

How may addition of fractions be proved? Ans. By subtracting one 
of the given fractions from the sum obtained, and seeing whether the re- 
mainder equals the sum of the remaining fractions. How may subtraction 
of fractions be proved? - Ans. By adding subtrahend and remainder, and 
seeing whether their sum equals the minuend. How may multiplication 
of fractions be proved? Ans. By dividing the product by the multiplier, 
and seeing whether the quotient equals the multiplicand. How may di- 
vision of fractions be proved? Ans. By multiplying divisor and quotient, 
and seeing whether their product equals the dividend. 


100 COMMON FRACTIONS. 


170. MisceLLANEous EXAMPLES. 


1. Find the sum, then the difference, and then the product of 
33 and 154. Divide 3% by 1y%. 
2. From a piece of cloth, $ and # of itself were cut off. What 


part remained ? Ans. 35. 
3. One third of a piece of cloth was cut off, and then 3 of 
What remained. How much of it was left? Ans. +45. 


4, A and Btogether have 1477 sheep, of which A owns #, and 
B #. How many belong to each? 

5. A person owning $ of a farm gives $ of his share to his 
sister, who divides it equally among her four sons. What part 
of the whole does each son receive? Ans. 34 

6. A owns 3% of a ship worth $15422; he sells B 3 of his 
share. What part of the whole does A then have? What part 
has B? What is the value of A’s part? Of B’s part? 


7. 88 is 44 of what number? 


88 is 123 then 1 twelfth is j of S8, or 8; and 12 twelfths, or the whole number, 
is 12 times §, or 96. Ans. 96.—We divide 8S by the numerator, and multiply by the 
denominator, 


8. 1200 is 22° of what number ? Ans. 2658. 
9. 1552 is +25, of what number? 76 is $2 of what number? 
10. 14042 is 44 of what number? 85 is 44 of what number? 


11. = of 1200 is 3 of what number? Ans. 1620. 
Find how much ,§, of 1200 is; then proceed as in Example 7. 
12. 4; of 1743 is 486 of what number? Ans. 418. 


13. 3 of 126 is 24 of what number? 

14. $2 of 8000 is 34 of what number? 

15. A farmer kept his sheep in two pastures; half of his flock 
was in one, and 87 sheep in the other. How many sheep had he? 

16. A sum of money is divided between A and B. A gets } 


of it, and B gets $360. How many dollars does A receive? 


If A gets 3, how many fourths does B get? If $360 is tree fourths, what will 
one fourth be? 


17. A sum of money is divided among A, B, and C. A gets 


4, B 4, and C $70. What was the amount divided? Ans. $168. 


How much do A and B together receive? What fraction is left for C? If &70 
equals this fraction, what will the whole be? 


MISCELLANEOUS EXAMPLES, 101 


18. How many sheep has a farmer, who keeps + of his flock 
in one field, 4 in a second, and the rest, numbering 779, in a 
third ? Ans. 1230 sheep. 

19, A merchant paid $272 for flour, at $93 a barrel. How 
many barrels did he buy ? 

20. What fraction is 885 of 6032 * Ans.: 325 = §. 

21. Paul has read 96 pages in a volume that contains 144 
pages. What fraction of the book has he read through ? 

22. A certain school is composed of 67 boys and 59 girls. 
What fraction of the whole do the boys form, and what the girls? 

23. Having a large cake, I divide half of it into five equal 
parts, and give three of these parts away. What portion of the 
whole cake have I left? Ans. 75. 

24. A lady divides $300 among her three sons, giving the 
first $75, the second $125, and the third the rest. What fraction 
of the whole does each receive? 

25. What number must be added to 43 to make 62? 

26. What number taken from 8} leaves 33? Ans. 444. 

27. The sum of two numbers is 4743; the less is 143. What 
is the greater ? 

28. Given, the quotient 227, the divisor 345,; required the 
dividend. Ans. 35. 
29. Subtrahend, 5%; remainder, 242; what is the minuend? 

30. The product of two factors is 132,9,5,; one of the factors 
is 12, what is the other? Ans. 117+. 

31. What is the difference between 37+ 3°; and # of 55? 

82. A bank paid out half of its money, then half of what re- 
mained, and again half of what then remained. What fraction 
“of the whole was left? Ans. 4 of 4 of $ 

33. A bank paid out 4 of its money, then 4 of what nateieee 
and again 4 of what then remained. What part of its money 
was left? Ans. 35;. 

34, C walks 3} miles an hour; D, 44. How many hours will 
it take C to walk 15 miles, and how many D? If they walk 
towards each other from two points 15 miles apart, how long 
before they will meet? Last ans. 2 hours- 


102 COMMON FRACTIONS, 


35. What number added to ~,+2;+32 will make 3? 

86. What number must be multiplied into 8 of 43 of 1432, to 
produce 38}? Ans. 34. 

37. F can mow a field in 8 hours, and G in 9 hours. What 
part can F do in one hour, and what part G? What part can 
both do in one hour? Ans. F, 4; G, 43; both, 44. 

38. P can dig a trench in 12 hours, Q in 15 hours. What 
part can each do in one hour, and what part both? 

39. A and B can mow a field in 14 hours; B alone can mow 
it in 24 hours; how long will it take A to do it? Ans. 333 hours. 


Iiow much can A and B together do in1 hour? How much can B alone do in 
J hour? How much is left for A to doin 1 hour? If he does this fraction in 1 hour, 
how long will it take him to do the whole? 


40. If A can dig a cellar in 20 hours, and B in 24, how long 
will it take both, working together, to do it? Ans. 1042 hours. 
41. Reduce ;', to ninety-sixths ($ 146). Ans. 42, 

42. How many forty-fourths in 7? In 3,4,? 

43. What is the smallest fraction which added to the sum of 
8° and $ will make the result a whole number ? 

44, A family consume 1% tons of coal in the parlor, 2% tons in 
the kitchen, and $ of a ton in each of their five bed-rooms. How 
much do they use in all? Ans. 744 tons. 

45. Four men agreed to share their earnings for one month 
equally. The first earned $604; the second, $408; the third, 
$50,3,; the fourth, $723%,. What did each receive? 

46. Will you increase or diminish the fraction ;%,, if you add 


4 to each of its terms, and how much? Ans. Inc. 4. 
47. Will you increase or diminish the fraction 43, if you sub- 
tract 2 from each term, and how much? Ans. Dim. 5. - 


48. Sold a house and lot for $42504. The house cost $37591; 
the lot, $8462. How much was gained or lost? Ans. $3564 lost. 

49, A man who has a journey of 78,5, miles to make, goes 
1542 miles the first day, and 23;; miles the next. How far has 
he then to go? Ans. 393735 miles. 

50. The difference between two fractions is ;’,; if the smaller 
fraction is ,3;, what is the greater? 


DECIMAL FRACTIONS. 103 


CHAPTER XI. 
DECIMAL FRACTIONS. 


171, A Decimal Fraction is one whose denominator is 
10, or 10 multiplied into itself one or more times. Its 
numerator only 1s written, with a dot (.) called the decz- 
mal point or separa'triz before it. Thus:— 


#s is written .9 $9. is written .889 
~i5 is written .11 ity is written .7777 


Decimal Fractions are briefly called Decimals. The 
term comes from the Latin word decem, ten. 

172. Decimals arise from successive divisions by ten. 
If a unit is divided into ten equal parts, each part is called 
one tenth. If one of these tenths is subdivided into ten 
equal parts, each of these subdivisions is one hundredth 
75 of 45 = zi,). So, from further divisions by 10, we 
get thousandihs, ten-thousandths, hundred-thousandths, 
millionths, &c. 


When tenths, hundredths, thousandths, &., are expressed with both 
numerator and denominator, they are common fractions; when with the 
numerator alone, preceded by a dot, they are decimals. As common frac- 
tions, they may be added, &c., according to the rules already given; but 
they are operated on much more easily as decimals. 


Notatien of Decimals. 


173. In writing integers, we found that the value of 
each figure depends on the place it occupies, being ten 
times as great as if it stood one place further to the right, 
and one tenth of what it would be in the next place to 
the left. Continuing this notation on the right of the 


171. What is a Decimal Fraction, and how isit written? Give examples. What 
are decimal fractions briefly called? What does the term dectmal come from?— 
172. Show how decimals arise. When are tenths, hundredths, thousandths, &c., 
common fractions, and when decimals? In whieh form are they most easily ope- 
rated on?—173, In writing integers, what did we find with respect to the value of 
each figure? Continuing this notation en the right of the units’ place, what do we 

ebtain ? 


104 DECIMAL FRACTIONS, 


t 
j 


units’ place, we obtain deeimal orders, each of which, 
as in the case of integers, invests its figure with a value 
ten times as great asthe next order on its right. 

Decimals are therefore expressed according to the 
same system as integers. Hence they may be written 
beside integers (with the separatrix to separate them), 
and may be added, subtracted, multiplied, and divided, 
.in the same way as integers. 
In the expression 2229,2222, each 2, whether integral or decimal, has 
a value ten times as great as the 2 next omits right. The mtegral twos 
represent collections of units ; the decimal twos, parts of a unit. 

174, Tante.—The names of the places on the right of 
the units’ place resemble those on the left. They may 
be learned from the following Table :— 


rs 
os SB wa 
of 3 ss we ees 
a = 2 got sles a. 

: ae Gs = ga Bes oe 
ay pope shee Ss & = ea = a de 
3 gi = a a = — 

rah sl 46s ms del ee Ae a 
S Soy 1 cel jie SoS Bw Ss og SoH ye S 
“= oso ~.g BUS 2 BS sya so 2S 

SU ten ae Eee tue Bs —SaSBEQBZURVPBERE4 & 
Seana B AR AS cho ve oa we See ue eee 
gees esedgVead ye stdtestyuserzyoas 

seiesgseeteaaodta eRe tog 84 a = 4 
@ ane Oi Big top ae aFsaog BRS FS 
HM HH ea MHP eR eD BRR R eae RRO 


ORDERS OF INTEGERS. ORDERS OF DECIMALS. 


Observe that in going from tens to hundreds, thousands, &e., we pass 
to higher orders; but in going from tenths to hundredths, thousandths, Xc., 
we pass to lower orders. 

Observe that two figures are required to express fens (10), one to ex- 
press tenths (.1); three for kandreds, two for heendredths; and generally, 
one less figure for a decimal order than for an order of integers of similar 
name. 


175. GENERAL PrincrpLes.—<As in whole numbers, so 
in decimals, we give a figure a certain value by writing 
it in a eertain place. Thus we express 


Where, then, may decimals be written, and how may they be added, &c. ?—1T4. 
What resemblance may be noticed in the names of the decimal orders? Name the 
orders of decimals, going to the right from the decimal point. Name the orders of 
integers, going to the Ieft. In which ease do we pass to higher orders, and in which 
to lower? How many figures are required, to express tens? To express tenths? 
To express hundreds? To express hundredths? What general principle is de. 
duced from this?—175. Ilow do we give a decimal figure s certain value? Give 
examples. 


NOTATION OF DECIMALS. 105 


ys by writing 9 in the place of tenths ee 
to by writing 9 in the place of hundredths .09; 
tooo by writing 9 in the place of thousandths .009, &c. 


176. From the above examples we see that vacant 
decimal places on the left must be filled with naughts. 
By leaving out the naughts in nine hundredths and nine 
thousandths, as written above, we would change them to 
nine tenths, 

177. We also see that to express a decimal we must use 
as many figures as there are naughts in its denominator. 
There is one naught in 10; one decimal figure expresses 
tenths. There are two nous hiae in 100; two Se fig- 
ures express hundredths, &c. 

178, It follows that every decimal has for its denomi-. 
nator 1 with as many naughts as there are figures in the 
numerator. 


179, A naught prefixed to a whole number does not 
change its value; every naught annexed multiplies it by 
10. With decimals it is not so. 

A naught prefixed to a decimal (on the right of the 
separatrix) throws its fi gures one place to the right, and 
thus divides it by 10: .3 is ten times as great as .03. 

A naught annexed to a decimal does not change its 
value, because denominator as well as numerator is mul- 
tiplied by 10. .8 = .80 (3, = 33%). 

180. Rutz.—7o express a decimal in figures, write its 
numerator as a whole number. If it contains fewer fig- 
ures than the denominator contains naughts, prefix naughts 
to supply the deficiency. Finally, prefix the decimal 
point. 

ExAMPLE.— Write forty-two millionths as a decimal. 


176. How must vacant decimal places on the left be filled?—177. How many 
figures must we use, to express a decimal?—178. What does every decimal have for 
its denominator ?—179. What is the effect of prefixing a naught to a whole number? 
To a decimal? What is the effect of annexing a naught toa whole number? Toa 
decimal ?—180. Recite the rule for expressing a decimal in figures, Give examples, 


106 DECIMAL FRACTIONS. 


Write the numerator as a whole number, 42. The denominator con- 
tains six naughts; hence, as the numerator contains but two figures, we 
must prefix to it four naughts. Ans. .000042. 


So, four and 357 millionths, 4.000857 
Ten and nineteen ten-thousandths, 10.0019 
Twenty and eighty-nine billionths, 20.000000089 


EXEROISE. 


1. How many figures are required, to express thousandths 

($177)? To express millionths? Billionths? Hundredths? Ten- 
thousandths? Hundred-trillionths? Ten-millionths? 

2. Give the denominators of the following decimals :— .001 ; 

.00001; .19; 4.1; .000003; 15.62; .83335; 5.162. 

3. Write the following as decimals, letting the decimal points 
range in line:—87 thousandths; 8 hundredths; 48 millionths; 
95 hundred-millionths; 490 hundred-thousandths; 1240 ten-mil- 
lionths; 10000004 hundred-millionths; 96 billionths; 9301 hun- 
dred-millionths; 2711 trillionths. 

4, Eight hundred and forty-one thousand ten-millionths. 

. Eighty-thousand,* four hundred and two millionths. 

. Seventy-one million three thousand and four billionths. 

. Eight hundred and ninety-six thousand hundred-millionths. 
. Forty-nine thousand,* and seven hundred-thousandths. 

. Sixty billion and fourteen thousand trillionths. 

10. Eight hundred million and ninety-nine ten-billionths. 

11. Seventeen thousand and forty-one ten-trillionths. 

12. 


Or 


eo mot ao 


0 vi e 7: s 3 a e e : . oe 
Toebo. % 109 T0009 Toecoon 9 Teo o) ahi: 
Numeration of Decimals. 


181, RuLte.—Read the numerator first as a whole num- 
ber, then name the denominator, as in common fractions. 


.09 is read Nine hundredths. 
.090018 Ninety-thousand and eighteen millionths. 
70.000000401 Seventy, and four hundred and one billionths. 


181. Recite the rule for reading decimals. 


* The comma is here used to show that what precedes it is whole number. 


NUMERATION OF DECIMALS, 107 


Exerrcist.—Read the following :— 


8 .010010101 46.0017 1.2 

90 .900909 3.123456789 463 

407 .00432002038 20.0020001 2.0308 

-6945 00076 1110111111 §.2381231123 
12003 -00002007 19.20200202 0.047 
.005007  .0509 5.0000004 .0004.000048 
36709 4000059 99.199099198 8.0019019 


182, Addition of Decimals, 
Ex.—Add 9.0421, .42386, 881, .03, and 23.5945. 


That we may unite things of the same kind, we 9.0421 
write the numbers down with their decimal points ‘ 


ranging perpendicularly in line, which brings figures 42386 
of the same order in the same column, Add as in 881. 

whole numbers, and place a decimal point in the re- .03 
sult under the points in the numbers added. 23.5945 


Rute.—l. Write the numbers with Ans 914.09046 
their decimal points ranging in line. 
Add as in whole numbers. Place the decimal point in 
the result under the points in the numbers added. 

2. Prove by adding from the top downward. 


EXAMPLES FOR PRAOTIOE. 


1, Add .128, .11496, 4.01, .06784, and 9.0342. Ans, 18.35. 

2. What is the value of .43974219.31+3.00067 + .0438851 
+5675.159 + 99.0004759 4+ 15006342 ? Ans, 5997.10376032. 

3. 8.84+450.329 4+ .988927 + 87.71-+.9 4.272073. 

A, .999 +999 + 9.887706 + .07809 + 88.199 +-.4. 

5. 7.714.853 49.6 +96 4.96 +.096 + 9604.54. 

6. 105.501 +4 0.105 + 8.648301 4.19 4+.776655432 + .3. 

7. Find the sum of 2063 millionths; 8064 ten-thousandths, 
$9 hundredths; 500, and 6009 hundred-thousandths; seven, and 
12 millionths; and 8638903 billionths. Ans, 508.359428003. 


18%. Set down the given cxample in addition of decimals, and perform the opera- 
sion. Give the rule. 


YOSs DECIMAL FRACTIONS. 


8. Add five thousandths; nineteen, and eighteen millionths; 
five hundred and twenty hundred-thousandths; forty, and seven 
tenths; 87 hundredths; 919 ten-thousandths, Ans. 60.672118. 

9. Required the sum of nineteen tenths; four hundred, and two 
hundredths; ninety-three thousandths; one hundred-thousandth. 

10. Add, as decimals, 83253 l0r 8503 xo'so! Qa ceso 


900003 Povo} 


183, Subtraction of Decimals, 


ExaMpLe.—F rom 4.19 subtract .000001. 


Write the subtrahend under the minuend, with 4.190000 
the decimal points in line. Write naughts in the vacant 009001 
places of the minuend (or supply them mentally), and pen abies 


subtract as in whole numbers. Place a decimal point Ans, 4.189999 
in the remainder under the other points. 


RuLE—1. Write the subtrahend under the minuend, 
with their decimal points ranging in line. Subtract as 
in whole numbers. Place the decimal point in the re- 
gnainder under the other decimal points. 

2. Prove by adding subtrahend and remainder. 


EXAMPLES FOR PRAOTICE. 


(1) (2) (3) (4) 
From 11. 8.00042 2.381400 23.56 
Take 897 875 _ 401006 1.0941875 
5. Subtract 47.99999 from 831.012. Ans. %83.01201. 
6. From .8754821 take .0006. Ans. .8748821. 
7. From 9.8 take the sum of .47 and 2.961. Ans. 5.869. 


8. Subtrahend, .88637; minuend, 312.42; required, the re- 
mainder. 

9. From one thousand take five thousandths. Ans. 999.995. 

10. Take 11 hundred-thousandths from 117 thousandths. 

11. From three million and one millionth, subtract one tenth. 

12, Find the value of 2.4+.609+.73 — 1.8. 

13. From eight and three tenths take eighty-four hundredths. 


183. Set down the given example in subtraction of decimals. Perform the opera- 
tion. Give the rule, 


MULTIPLICATION OF DECIMALS, 109 


14. From 83.1 subtract .8176; from the remainder take +32,, 
15. Find the difference between -82,, and 4;43755, first as com 
mon fractions, then as decimals. Do the results agree ? 


184, Multiplication of Decimals, 


Exampre 1.—Multiply .324 by .03, 


Write the given decimals as common fractions, and 6324 
multiply. +324; x 730 = 7itG%ia, which expressed deci- 03 
mally is .00972. The same result is obtained by multi- ee 
plying the given decimals together, and prefixing two Ans, .00972 
naughts and the decimal point to the product. 


Why prefix two naughts ?—The multiplicand containing 3 figures, its 
denominator contains 3 naughts ($178). The multiplier containing 2 
figures, its denominator contains 2 naughts. Hence the product of their 
denominators contains 3 +2 naughts; and the product of their numerators 
must contain 3 +2 figures ($177). As it has but three figures, we prefix 
two naughts.—The product of two decimals must therefore contain as 
many decimal places as both factors contain. 

Ruie.—1. Multiply as in whole numbers. From the 
right of the product point off as many figures for deci- 
mals as there are decimal places in both factors. If there 
are not so many, prefix naughts to supply the deficiency. 

2. Prove by multiplying multiplier by multiplicand. 


ExampLe 2.—Multiply 3.8 by .97. 


Multiply as in whole numbers. There : 
being 1 decimal place in the multipli- pe Proof .94 
cand, and 2 in the multiplier, point off 97 3.8 
1+ 2, or 8, figures from the right of the 266 776 
product. 342 291 
185. To multiply a decimal jns 3.686 3.686 


by 10, 100, 1000, &c., remove 
the decimal point as many places to the right as there are 
naughis in the multiplier. If there are not figures enough 
for this, annex naughts to supply the deficiency. 

015 x 10 = b5 


.015 x 1000. = 15 
015 x 10000 = 150 
184, Multiply .324 by .03 in the two ways shown above. Why do we prefix two 
naughts to the decimal product? KRecite the rule for the multiplication of decimals, 
—185, How may we multiply a decimal by 10, 100, 1000, &c.? Give examples, 


110 DECIMAL FRACTIONS. 


EXAMPLES FOR PRAOTIOE. 


(1) (2) (3) (4) 
Multiply 81.009 008765 7.91365 3256.9 
By 4,067 .0495 8.401 4.0008 


5. Multiply together 6.321, .987, and 1000. Ans, 6238,827, 

6. Multiply .4639721 by .00832. Ans, .003860247872, 

7. Multiply 5.482 by 21; by .21; by 9.8; by .00008. 

8. Multiply by 100 the following: .1; .003; .00007; 1.14. 

9, Find the product of one billionth and one billion. 

10. Multiply 73 thousandths by 19 hundredths. 

11. Multiply ninety-seven millionths by ten thousand. 

12. Multiply .00468 by 3.0009. Multiply 14.7 by .0908006. 

13. Find the product of 735, 7%, and 3%), first as common 
fractions, then as decimals. Do the results agree ? 

14, Multiply the sum of nineteen hundredths and eighteen 


thousandths by seventeen ten-thousandths. Ans. .0003536. 
15. Multiply the difference between two ten-thousandths and 
two hundred-thousandths by nine tenths. Ans. .000162. 


Division of Decimals. 


186. Division is the converse of multiplication. The 
dividend corresponds with the product, the divisor and 
quotient with the factors. 

Now, in multiplication of decimals, we found that the 
product contains as many decimal places as both factors 
together. Hence, in division of decimals, the dividend 
must contain as many decimal places as divisor and quo- 
tient together; and the quotient, as many as the decimal 
places in the dividend exceed those m the divisor. 

187, Rutze.—1. Divide as in whole numbers. Point 
of from the right of the quotient as many figures as the 
decimal places in the dividend exceed those in the divisor. 


186, In division of decimals, how many decimal places must the dividend con- 
tain? How many must the quotient contain? How does this follow from the mode of 
- pointing in multiplication of decimals ?—187. Give the rule for division of decimals 


DIVISION OF DECIMALS, lil 


Lf there are not so many, prefix naughts to supply the 
deficiency. | 

2. Prove by multiplying divisor by quotient. 

ExamMpLeE 1.—Divide 84.0065 by .05. 

Divide as in whole numbers. There being 4 deci- 
mal places in the dividend, and 2 in the divisor, point -05) 84.0065 
off 4— 2, that is 2, figures from the right of the quo- Ans, 1680.13 
tient. 

188, Annexing naughts to a decimal does not change 
its value. Hence, when the dividend contains fewer deci-_ 
mal places than the divisor, annex naughts to it till its 
decimal places equal those of the divisor ; then divide, 
and the quotient will be a whole number. 

ExAMPLE 2.—Divide 7240.5 by .0009. 


Geel cclices Seal ase: of tte Ge Aree -0009) 7240.5000 
quotient is a whole number. Ans. 8045000 

189. When there is a remainder, after using all the 
figures of the dividend, naughts may be annexed to the 
dividend and the division continued. Jn pointing off the 
quotient, these naughts must be counted as decimal figures 
of the dividend. The sign + is annexed to a quotient, 
to show that the division does not terminate. 


EXAMPLE 3.—Divide .075 by 4.3. 


00 
4.3) 075 (174 After using all the figures of the dividend, we 
43 annex naughts (placed above it), to continue the divi- 
320 sion, which may thus be carried out as far as desired. 
30] Using 2 naughts, we have 5 decimal places in the 
ee dividend, and 1 in the divisor. We must therefore 
190 point off 5—1, or 4, figures from the right of the que- 
172 tient, which requires us to prefix to it a naught. 
Sere Ans. .0174+ 
190. To divide a decimal by 10, 100, 
Ans, 0174+ 


1000, &c., remove the decimal point as 


188. When the divisor contains more decimal places than the dividend, how 
must we proceed? Divide 7240.5 by .0009.—189. When there is a remainder after 
using all the figures of the dividend, what may be done? In pointing off, how must 
we consider these annexed naughts? What does the sign + annexed toa quotient 
show? ‘Apply this rule in Ex, 3.—190. How may we divide a decimal by 10, 10Q 
1000, &e.? Give examples. 


112 DECIMAL FRACTIONS. 


many places to the left as there are naughts in the divisor. 
If there are not. figures enough for this, prefix naughts 
to supply the deficiency. 


156.3 — 10 = 15.63 
156.8 + 1000 = .15638 
156.8 ~ 100000 = .001563 


EXAMPLES FOR PRAOTICE. 


Find the value of the following; prove each example :— 


1. 144+ .36 Ans. .4| 11. Divide .00063 by 9. 

2. .49 + 700 Ans. .0007 12. Divide .5 by 50000. 

3. 1382+.11 Ans. 1200. 18. Divide .491 by .00007. 
4, .8+17.3 Ans. .109589+ 14. Divide 810 by .000009. 
5. Divide .75 by .7500. 15. Divide .0001 by .001. 
6. Divide 10000 by .01. 16. Divide 2880 by .0036. 
7. Divide .24 by 60. 17. Divide 19 by 42.96. 

8. Divide 8487 by 1.8. 18. Divide .120 by 100000. 
9, Divide 1210 by .11. 19. Divide 64000 by .0016. 
10. Divide .00001 by 1001. 20. Divide 34%; by wine: 


21. Divide 639.521 by 10000; by 100; by 10000000; by 10. 
22. Divide one million by one ten-thousandth. 
23. Divide the sum of 941 thousandths and 38 hundredths by 


one thousand, Ans. .001821 
24. Divide the difference between eight tenths and one mil- 
lionth by seventy-nine hundredths. Ans. 1.0126569 + 


25. Divide the product of one hundredth and one thousandth 
by one ten-billionth. 

26. Divide 7 tenths by 3. es and the quotient by 20000. 

27. Divide 74532, by 744), first as common fractions, then as 
decimals. Do the results aihced ? 

28. Divide the sum of five thousand and two thousandths by 


two hundredths. Ans. 250000.1 
29. Divide the difference between 200 and 2 hundredths by 
9 hundredths. Ans, 2222. 


30. Divide by 100 the following: 26.83; .2683; 268.3; .02683. 


REDUCTION OF DECIMALS. 113 


Reduction of Decimals, 


191, Case I.—7o reduce a decimal to a common frac- 
tion. 

Ruie.— Write the given decimal, with its point omitted, 
over its denominator, and reduce this common fraction 
to its lowest terms. 


. EXAMPLE.—Reduce .125 to a common fraction. 
195 = 725, = 4 Ans. 
EXAMPLES FOR PRACTIOE. 


Reduce the following to common fractions:— 


Wee. 870 5 1b 4425-88. 7. 46; .046; .0046; .11. 

2. .225; .485; 575; .656, 8. .076; .075; .0075 138. 
8. .00375 Ans. s$5.| 9. .00764 Ans, xi84,. 
4, .000225 Ans. zshop- | 10. .00025 Ans. xis: 
5, .386984 Ans. ~8235.| 11. .000155 Ans. xphbon- 
6. .0982 Ans. 2d;.| 12. .01250505 Ans. ~250104,. 


192. Casz IL—Zo reduce a common fraction to a 
decimal, 


ExampLe.—Reduce 4 to a decimal. 
x is 1 divided by 8. To perform the division, annex 


decimal naughts to the dividend 1, and divide by 8. Point 8) 1.000 
off three figures from the right of the quotient, because _ ep mteg a= 
there are three decimal places im the dividend and none Ans, .125 


in the divisor. Ans. .125. 

To prove the result, reduce .125 back toa common fraction ($ 191), 
and see whether it produces 4 ae 

Rutz.—1. Annex decimal naughts to the numerator, 
and divide by the denominator. Point off the quotient 
as in division of decimals. 

2. Prove by reducing the decimal obtained back to a 
common fraction. 

Compound fractions must first be reduced to simple ones. 

191. Recite the rule for reducing a decimal to a common fraction, Give an ex- 


ample.—192. Solve and explain the given example. Recite the rule for reducing a 
eommon fraction to a decimal, What must first be done with compound fractions? 


114 DECIMAL FRACTIONS. 


EXAMPLES FOR PRAOTIOERN. 


Find the value of the following in decimals :— 


1.33 25 $3 $3 wos Do 8. $3 83 fo3 so3 Bho 

2. P53 1853 s23 $2 9 203 43 F053 Dee 

3. 4 of 2 of f. Ans. .225 | 10.4—3,. Ans. .0T 
4, 4 of 2 of § of 14. 11.444. Ans. .95 
5. soffof4. Ans. .015625 12.$4+4+7;,. Ans. .9875 
6. ¢ of 4 of 1. Ans. .008 | 138. 33 + 7. Ans. .625 
T &xpox4ex4h. Ans..838+ | 14. 2+ 2%. Ans. 4545+ 


Circulating Becimais, 


193, Sometimes (as in Example 7 and 14 above) no 
decimal can be obtained exactly equivalent to a common 
fraction. This is because the division does not terminate, 
but the same figure or set of figures keeps recurring in 
the quotient. In such cases, the further the division is 
carried out, the more nearly correct will the answer be. 

194, A decimal in which one or more figures are con- 
stantly repeated, is called a Circulating Decimal. The 
repeated figure or figures are called the Repetend. 

195, A repetend is denoted by a dot placed over it, if 
it is a single figure,—or over its first and last figure, if it 
contains more than one. Thus: 3 .833, &e 45 = 
454545, &e. .2148148, &c. 

196, A Pure Circulating Decimal is one that consists 
wholly of a repetend; as, .3, .243. 

A Mixed Circulating Decimal is one in which the repe- 
tend is preceded by one or more decimal figures, which 
form what is called the Finite Part: as, .23; .2 is the 
finite part. 


193. Why, in some cases, can not a decimal be obtained equivalent to a common 
fraction?—194, What is a Circulating Decimal? What are the repeated figure or 
figures called ?—195, Mow is a repetend denoted ?—196. What is a Pure Circulating 
Decimal? What is a Mixed Circulating Decimal? Give examples, 


CIRCULATING DECIMALS. 115 


197. Repuctrion or CircuLtating Drecimats.—Reduc- 
according to the rule in § 192, we find 


ing 
4 = 111111, &. or, .i Hence, reasoning backward, .1 = } 
% = .222229, &e. or, .2 “ « “ 2 = 2 
2 = .883333, &c. or, .3 cs ss x 32 
gy = .010101, &e. or, .01 “ “ «“ Ol = x 
gs = .020202, &e. or, .02 v6 «“ a 02 = % 
gs = .030308, &. or, .03 es a “ 03 = ¥& 
atx = .001001, &c. or, 0Oi « - c 001 = xhy 


It will be seen from the above that the denominator 
of a repetend consists of as many nines as it contains 
figures. Hence the following rule :— | 

198. Route I.—7o reduce arepetend to a common frac- 
tion, write under it for a denominator as many nines as 
tt contains figures. : 

199. Rute Il.—7Zo reduce a mixed circulating deci- 
mal, reduce the repetend to a common fraction, as above, 
annex it to the finite part, and place the whole over the 
denominator of the finite part. Reduce the complex frac- 

tion thus formed to a sinuple one. 


Exampite. Reduce .2336 to a common fraction. 


Reduce 36 to a common fraction : ee 

Annex the fraction to the finite part: » 23-4 
23-4 

Place the whole over the denom. of the finite part: ie 

Reduce the complex fraction thus formed: / Posi; Ans. 


EXAMPLES FOR PRACTICE. 


1. Write as circulating decimals: .8333+ (.8); .263263263-+ 
(.263); .10471047+ ; .246666+4 (.246); 14921214; 4.9871871+4 ; 
8.2300300+ ; .12345671234567+ ; 2494+. 


197. Of what does the denominator of a repetend consist? How is this shown? 
—198. Recite the rule for reducing a repetend to a common fraction.—199. Recite 
the rule for reducing a mixed circulating decimal to a common fraction, Apply this 
rule in the given example. : 


116 FEDERAL MONEY, 


2. Write as circulating decimals ($197): 4 (.7)3 <5 (.07); v3 
(007); se'ou3 vyda03 $93 vo0es odb05 cre 'coss $3 ves vhs 
35643 


ovoos 435 30835 $5882. 
3. Reduce to common fractions ($198) in their lowest terms: 
25; .081; .815; .1000. Ans. 353 z3x3 Bri $999. 
4. Reduce to common fractions ($199) in their lowest terms: 
06; .243; .68219; .8i003. Ans. Js; Jas3 $89%5 43492. 

5. How much more is .8 than .8? Ans. 45. 

6. How much more is .21 than .21? Ans. xz55- 

7. How much less is .72 than .72? 

8. Which is the greater, .48 or .48, and how much? 

9. Reduce .128. Ans. Ajy.| 18. Reduce .12 and .1894. 
10. Reduce .321. Ans. 49%.| 14. Reduce .083 and .4896. 
11. Reduce .2763. Ans. sh8.| 15. Reduce .135 and .0398. 
12. Reduce .0045. Ans. x$y.! 16. Reduce .135 and .6345. 


CHAPTER XII. 
FEDERAL MONEY. 


200. A Coin is a stamped piece of metal used as money. 

201. By the Currency of a country is meant its money, 
consisting of coins, bank bills, government notes, &c. 

202. Different countries have different currencies. The 
currency of the United States is called Federal Money. 


TasLE oF FrepERAL Money. 
10 mills (m.) make 1 cent,... ¢., ct. 


10 cents, Tedimey esac. 
10 dimes, 1 dollar,. . $ 
10 dollars, 1 eagle, .. EH. 


200, What is a Coin?—201. What is meant by the Currency of a country ?—202. 
What is the currency of the United States called? Recite the Table of Federal Money. 


UNITED STATES COINS. *1ULZT 


\ 


The mill, one thousandth part of a dollar, takes its name from the 
Latin word mille, a thousand; the cent, one hundredth of a dollar, from 
the Latin centwm, a hundred ; "the dime, one tenth of a dollar, from the 
French dime, a tithe or tenth. The word dollar comes from the German 
thaler. The dollar-mark § is supposed to have originated from the letters 
U.S. (for United States) written one upon the other. 


203. Unirep States Corns.—The coins of the United 
States represent all the denominations of the above Table 
except mills, as well as other values. They are as fol- 


lows :— 

Gotp. Double eagle, worth $20. | Smtver. Dollar, worth $1. 
Eagle, “ — $10. Half-dollar, i OO Cs 
Half-eagle, Ov BaD. Quarter-dollar, “ 25¢. 
Three-dollar piece, “ § 3. Dime, SLO 
Quarter-eagle, beats aes Half-dime, ‘Serna Cs 
Dollar, ane Three-cent piece, “ 8c, 


Copper AND NicxeL. 5c. and 3c. piece (2 copper, } nickel). 
Copper. 2c. piece (no one coined). 
Cent (425 copper, ra tin and zinc). 


The gold and silver coins are nine tenths pure metal, the former being 
alloyed with one tenth of silver and copper, and the latter with one tenth 
of copper. The silver half dime and 8c. piece are no longer coined. 

204. Writing AND Reapinc FepErrRAL Monry.—In 
passing from mills to cents, from cents to dimes, and 
from dimes to dollars, we go each time to a denomina- 
tion ten times greater, just as we do in passing from 
thousandths to hundredths, from hundredths to tenths. 
Federal Money is therefore a decimal currency, and may 
be written and operated on in all respects like decimals. 

205. In writing and reading Federal Money, the only 
denominations used are dollars, cents, and mills. The 
dollar is the unit or integer, and is separated by the deci- 


From what does the mill take its name? The cent? Thedime? The dollar? 
How is the dollar-mark supposed to have originated ?—203. Name the gold coins of 
the United States, and their value. The silyer coins. The copper coins. What pro- 
portion of the gold and silver coins is pure metal? With what are they alloyed? Of 
what does the cent consist ?—204. What kind ofa currency is Federal Money? How 
may it be written and operated on?—205. What denominations are used in writing 
and reading Federal Money? Which of these is the integer? How is it separated 
from cents ? 


118 ° FEDERAL MONEY. 


mal point from cents, which occupy the first two places 
on the right of the point, mills occupying the third. 
Hence the rules. 

Cents occupy two places,—that of dimes and their own,—because we 
do not recognize dimes in reading. Cents are sometimes written in the 
form of a common fraction, as hundredths of a dollar; as, $5;7j5- 

906. Rute l— Write Federal Money decimally, the 
dollars as integer, the cents as hundredths, the mills as 
thousandths. 


ExampLes.—Twelve dollars, six cents, $12.06 
Twelve dollars, sixty cents, $12.60 
Twelve dollars, six cents, five mills, $12.065 
Twelve dollars, five mills, $12.005 


207. Rute I.—m reading Federal Money, call the 
snteger dollars, the hundredths cents, the thousandths mills. 


EXAMPLES FOR PRACTICE. 


resy 


. Write eleven dollars, eleven cents. 

. Write six hundred dollars, three mills. 

. Write ninety-eight dollars, seven cents. 

. Write one thousand dollars, ten cents, nine mills. 
. Write six dollars, seventeen cents, eight mills. 

6. Write ninety-nine cents, nine mills. 

7. Write a million dollars, one cent, one mill. 


H= G2 bo 


| 


8. Read 840.268 $560.005 $ .009 
$923.01 $5.90 - $1.00 
$14296.30 $980.09 $11.111 


208. OprRATIONS In Feprrat Monry.—7o add, sub- 
tract, multiply, or divide Kederal Money, express the given 
amounts decitmally, and proceed as in decimals. 


Represent 4 cent as 5 mills, + cent as 25 ten thousandths of a dollar. 
Thus: 3874 cents = $.3875 61 cents = $.0625 


How many places do cents occupy? Why? Howare cents sometimes written? 
—206. Recite the rule for writing Federal Money.—207. Recite the rule for reading 
Federal Money.—208. Give the rule for adding, subtracting, multiplying, or dividing 
Federal Money. How is } cent represented? 4 cent? 


OPERATIONS IN FEDERAL MONEY. 119 


As there are no mills coined, less than 5 mills in a result is dis- 
regarded in business dealings, and 5 mills or more are called an additional 
cent. 


ExampLE 1.—Add together $4.83, $4.83 
$10.005, $480, $.374, and $3.36. 10.005 
Follow the rule for the addition of decimals, 480.00 | 
§ 182. Write the items with their decimal points 379 
in ae representing the half cent as 5 mills. Add, 3.36 
and bring down the decimal point under the points $. 
in the items added. Ans. $498.57. ° Ans, $498.570 
$14.00 ExaMPLe 2.—A person having $14 spent 
9.83 $9.83 ; how much had he left ? 
8 417 Ans. He had left the difference between $14 and $9.83. 


Proceed as in subtraction of decimals, § 183. Ans. $4.17. 


Exampte 3.—What will 12 coats cost, at $14.75 each? 


If 1 coat costs $14.75, 12 coats will cost 12 times 814,75 
$14.75. Proceed as in multiplication of decimals, . 
$184. Point off two figures at the right of the prod- 12 
uct, because there are two decimal places in multipli- 4,. $177.00 
cand and multiplier. Ans. $177. 

Exampite 4.—How many photographs, at 124 cents 
apiece, can be bought for $6.25 ? 


As many as 12} cents is contained times in $6.25. 
-125)6.250(50 — Proceed as in division of decimals, $187. Annex a 
625 naught to the dividend, to make its decimal places 
Se equal those of the divisor, and the quotient will be 

00 a whole number. -Ans, 50 photographs. 


EXAMPLES FOR PRACTICE. 


1. Add together forty-three dollars; seven dollars, twenty 
cents; nineteen cents, nine mills; twenty dollars, three mills; 
four cents, six mills; and seventy-five cents. Ans. $71.198. 

2. Take forty-three cents from thirty dollars. 

3. From nine dollars, nine cents, subtract eight dollars, eighty 
cents, eight mills, 

4, A farmer received $41.60 for poultry, $125 for a horse, 
$3.12 for eggs, and $5.55 for cheese. What was the sum total? 


’ Tfow is less than 5 mills in a result regarded? How are 5 mills or more re- 
garded? 


120 FEDERAL MONEY. 


5. A father divides twenty thousand dollars equally among his 
7 children; how much does each get? Ans. $2857.142 + 


To get cents and mills in the answer, annex decimal naughts to the dividend, 
and continue the division. 


6. How much is four and a half times $13.13 ? 


In such a case express the fraction of the multiplier decimally. $13.18 x 4.5 
7. If 23 acres are worth $724.50, how much is # of an acre 
worth? Ans. $23.625. 


First find how much 1 acre is worth; then £ of an aere. 

8. A owes B $75.93, and borrows of him $37.50 more. If A 
then pays B $100.75, how much will still remain due? 

9. What cost 185 pounds of coffee, at $.298 a pound ? 

10. Bought 8.875 cords of wood, at $5.50 a cord. What did 
it cost? Ans. $46.06. 

11. The Erie Canal is 368 miles long, and cost $7148789. 
What was the average expense per mile? : 

12. A farmer sold his butter for 27 cents a pound, receiving 
$982.935. How many pounds did he sell? Ans. 8640.5 pounds. 

18. What cost 16 sofas, at $43.75 apiece ? 

14. The butter made in one year from the milk of 58 cows, 
having been sold for 80 cents a pound, brought $2369.10. How 
many pounds were sold, and what was the average amount pro- 
duced by each cow? Ans, Average, 149 pounds. 

15. A man having 7 sons and 4 daughters, divides $100 among 
his sons, and $75 among his daughters. By how much does each 


daughter’s share exceed each son’s share? Ans. $4.463. 


Tow much is each son’s share (+ of $100)? How much is each daughter’s share 
(¢ of $75)? Find the difference between these two amounts. 


16. How much will a man waste on segars in 50 years, if he 
smokes four daily, averaging 44 cents each, allowing 865 days to 
the year? Ans. $8285. 

17. A person who earns $1050 a year, spends in January, 
$98.41; in February, $81.38; in March, $102.28; in April, 
$125.26; in May, $74.88; in June, $73.47; in July, $65.98; in 
August, $87.21; in September, $70.84; in October, $122.08; in 
November, $79.68; in December, $52.77. How much has he left 
at the end of the year? Ans. $16.81. 


a 


EXAMPLES FOR PRACTICE. 121 


18. How many pounds of cheese will be made from 46 cows 
in 40 days, if each cow averages 2.5 gallons of milk daily, and 
each gallon produces 1.1 pounds of cheese? What will the whole 
bring at 184 cents a pound? Ans. $702,075. 


How much milk is produced daily by 46 cows? How much cheese is produced 
daily? How much cheese, then, is produced in 30 days? What will this bring at 
$.185 a pound ? \ 


19. If I lay in eleven tons of coal at $9.75 a ton, two barrels 
of charcoal at 95 cents a barrel, and three loads of wood at $4.25 
a load, and pay $3.80 for sawing and splitting, what does my 
fuel cost me? Ans. $125.70. 

20. Suppose that a man buys two glasses of liquor.a day, at 
ten cents a glass; how many volumes costing $1.50 each, could 
he purchase with the sum that he would thus spend in 30 years, 
allowing 865 days to the year? Ans. 1460 volumes. 

21. A farmer buys 23} (28.25) yards of cloth at $3.75 a yard ; 

if he pays for it with butter at 80 cents a pound, how much butter 
must he give ? Ans. 290.625 pounds. 

22. What will 24 copy-books cost, at 12 cents apiece ? 

23. The expenses of a family for May are as follows :—fuel, 
$10.25; table, $47.90; clothing, $13; rent, $31.25; sundries, 
$9.58. The next month they diminish their expenses one half; 
what does it cost them to live in June? Ans. $55.965. 

24, A ferry-master who received $5.26 one morning, and $7.93 
in the afternoon, found that he had taken a counterfeit dollar-bill 
and two bad quarter-dollars. How much good money did he 
take that day ? Ans. $11.69. 

25. The Welland Canal, 36 miles long, cost $7000000, What 
was the average cost per mile? 

26. On the debtor (Dr.) side of an account are the following 
items: $1050, $241.71, $99.88, $760, $437.75. On the creditor 
(Cr.) side are the following: $69.95, $860, $.875, $48.17. What 
is the balance ? 


The balance is found by adding the items on the debtor side, then those on the 
* sreditor side, and taking the less sum from the greater. 


27. What is the value of 42 bales of cotton, containing 425} 
pounds each, at 56 cents a pound? Ans. $10007.76. 
oa 


122 FEDERAL MONEY. 


28. A farmer exchanges 9 tons of hay, worth $33.75 a ton, for 
wheat at $2.10 a bushel. How many bushels should he receive ? 

29. What cost 144 paper-cutters, at 374 cents each? 

30. Three partners bought some land for $9375. They sold 
it for $1100 cash, $973.50 worth of produce, and notes to the 
amount of $8000. What was each partner’s profit? Ans. $232.834. 


How much did they receive for the land in all—cash, produce, and notes? What 
was the whole profit? Divide this among three. 


31. C and D bought 380 acres of land each. CO sold his so as 
to gain $1.75 an acre. D sold hissoas to gain four times as much 
as C. How much did D make on his 30 acres? 

How much did D make on 1 acre? How much on 30 acres? 

32. A man buys 9 chairs, at $2.75 each. He sells them ata 
profit of 50c.each. Whatdoeshe get forthe whole? Ans. $29.25. 

33. The profits of a certain firm for one year are $8961. One 
of the partners, who receives 4 of the profits, divides his share 
equally among his five sons. How much does each son receive? 

34, If I buy some lace for $2.624 a yard, and sell it for $3.10, 
do I gain or lose, and how much ? 

35. If I receive $3.15 a barrel for apples that cost me $3.874, 
do I gain or lose, and how much ? 

36. F’s account at a certain store stands as follows :—Debits, 
$49.75, $63, $3.75, $15.875, $304.05, $27, $199.875. Credits, 
$415, $88.80, $42. . What is the balance? Ans. $117. 

87. Find the balance of each of the two following accounts; 


—— 


Dr. Cr. Dr. Cr. 
$84.09 $149.68 $9264.00 . $1860.43 
63.86 19.94 185.76  10249.75 
69.88 286.41 12458.68 76.49 
2726.45 59.88 5289.21 3486.91 
765.50 1147.31 18.75 590.43 
338.88 6.66 451.39 1751.52 
47.67 999.88 865.56 8064.63 
4827.33 1428.72 4157.88 591.27 
695.63 2379.64. 886.27 7208.48 
5820.94 822.56 21934.33 2457.87 °°" 
967.19 1865.63 13642.85 8599.19 


Ane. Balance $7241.11 


Ans. Balance $21022.66 


ee 


ALIQUOT PARTS or $1. £233 


209. Ariquot Parts.—Aliquot Parts of a number are 
either whole or mixed numbers that will divide it without 
remainder, 44, 3, and 24, are aliquot parts of 9. 


When an aliquot part is a whole number, it is a factor; 3 is both an 
aliquot part and a factor of 9. 


210. The aliquot parts of a dollar most frequently 
‘used, are as follows :— 


50 cents = 4 of $1. 124 cents = 4 of $1. 
834 cents = 4 of $1. 10 cents = +5 of $1. 
25 cents = 4 of $1. 64 cents = + of $1. 
20 cents = ¢ of $1. 5 cents = x5 of $1. 


211, When the cost of anumber of articles is required, 
and the price of one is an aliquot part of $1, we may save 
work by operating with it as the fraction of $1. 

Ex.—What will 600 Spellers cost, at 331 cents each? 


At $1 each, 600 Spellers would cost 
$600. But 334 cents are 5 of $1; there- 4 of 0- = 
fore: at 834 cents, they will cost ,0f. .3 of $60 $200 Ans. 
$600, or $200. 

Ruitz.— Take such a part of the given number as thé 


price is of $1, and the result will be the answer in dollars. 


212. In like manner, to divide by an aliquot part of a 
dollar, divide by the fraction that represents it. 


ExamMpPLe.—How many pass-books, at 61 cents each, 
ean be bought for $2 ? 


As many as 64 cents is con- ie 
tained times in $2. 64 cents is 6} cents = Sy 
yy ofa dollar. =); of $1 is con- 
tained in $2 82 times. Ans.32 2+ 74 = 2 x 18 = 32 Anz. 
pass-books. 


EXAMPLES FOR PRAOTICE. 


1. What cost 144 pencils, at 64 cents each? Ans. $9. 
2. What cost 500 Primers, at 20 cents each ? Ans. $100. 


209. What is meant by Aliquot Parts? When is an aliquot part a factor, and © 
when not?—210. Mention the aliquot parts of a dollar most frequently used.—211. 
What is the rule for finding the cost of a number of articles, when the price of one is 
an aliquot part of $1?—212. What is the rule for dividing by an aliquot part of $1? 


FEDERAL MONEY. 


—— 
ho 
1em 


. What cost 1600 pounds of sugar, at 124 cents ? 

. What cost 1728 bottles of ink, at 124 cents? 

At 834 cents a pound, what cost 150 pounds of candles ¢ 

. What cost 2500 pounds of soap, at 124.2? Ans. $312.50. 
. What must I give for 85 rulers, at 50 cents each ? 

. At 64 cents apiece, what cost 1000 cabbages? 

. If 1 yard of muslin costs 834 cents, what cost 96 yards ? 
10. How many mackerel at 334 cents, can be bought for $15? 
11. How many dozen eggs, at 25 c. a dozen, can I buy for $6? 
12. How much sugar, at 10. a pound, can be bought for $20? 
13. How many baskets of berries, at 64 ¢., will $12 buy? 

14. At 124 ¢. a quart, how many quarts of nuts will $5 buy? 


© CO To Oo # oO 


213, ARTICLES SOLD BY THE 100 or 1000,—The price 
of articles is sometimes given by the 100 or 1000. 


ExamMpLe.—W hat will 1550 envelopes 1550 
cost, at $3.25 a thousand ? 3.25 

If we multiply the number of articles and the “750 
price per thousand together, we get a result 1000 3100 


times too great. We must therefore divide the prod- 4650 
uct by 1000,—that is, point off three addtional fig- oo ae 
ures. Hence the rule:— ' 5037.50 
Rutz.— 70 find the cost of a number — Ans. 5.0375 

of articles whose price is given by the 

100 or 1000, meliiply the given number and price together, 
and point off two additional figures in the product tf the 
price is per hundred, or three if it is per thousand. 


The price of lumber (boards, plank, logs, &e.) is generally given by 
the thousand feet. Per CO. (for centum) means a hundred ; per W. (for 
mille) means a thousand, 


EXAMPLES FOR PRAOTICE. 


1. What cost 525 oysters, at 95 c. a hundred? Ans, $4.9875. 
2. At $4.20 a thousand, what cost 5625 envelopes? 
3. What cost 1750 pounds of dried codfish, at $7.15 a hundred ? 


213. Solve the given example. Recite the rule for finding the cost of a number 
of articles whose price is given by the 100 or 1000. How is the price of lumber gene- 
rally given? Wat docs per @. mean? Per IL? 


MAKING OUT BILLS. 125 


. What must I pay for 1300 feet of boards, at $29 per M.? 
. What cost 425 feet of white-oak logs, at $65 per M.? 
. At $10.50 a hundred, what cost 9870 pounds of lead ? 
What cost 5000 laths, at 45 cents per C. ? 
. What is the freight on 962 pounds, at $1.25 per 100? 
. What must be paid for laying 1275 bricks, at $8 per 1000? 
10. Required the cost of 90422 bricks, at $7.75 a thousand ? 
11. How much must be paid for planing 4976280 feet of boards, 
at 84 cents per 1000 feet ? Ans. $4180.075. 
12. Ifa man carts 575 loads of bricks, averaging 1800 to the 
load, and is paid at the rate of 95 c. a thousand, how much will 
he receive ? Ans. $983.25. 

18. Sold, at $114 a hundred, three cargoes of pine-apples, the 
first consisting of 840, the second of 970, the third of 724. How 
much did they all bring? Ans. $285.075. 

14. Bought 2000 feet of boards, at $27 per M., and 675 feet of 
pine stuff, at $3.80 per C.- What was the whole cost? Ans. $79.65. 


SOT A te 


Making out Hills. 


214, A Bill is a statement of what one party owes 
another for goods bought or services rendered. It may 
consist. of several items, which are added or “footed up”, 
to find the whole amount. Specimens of bills follow. 


215, On the first line (see Bill 1, page 126) stand the name of the 
place and the date. On the second line is the name of the party owing the 
bill, and on the third that of the party to whom itisowed. To ce 
means To Debtor, or indebted. Bt of , which is another 
form, used in Bill 2, means Bought of 

Then come the items, each with its date if the dates are different, as in 
Bill 4. @ before the price means af. 

When a bill is presented and not paid, it is left without signature, like 
Bill 1. When it is paid, the party receiving it signs his name under the 
words feceived Payment, as in Bill 2, and the person paying it retains it 
as evidence that it is paid. <A clerk or collector signs his name for his 
principal, as shown in Bills 3 and 5 


214, What is a Bill?—215. What is found on the first line of a bill? On the 
second line? On the third line? What is meant by Zo —— Dr.? Btof? What 
is the meaning of the sign @ in the items? When is a bill left without signature ? 
When a bill is paid, what does the party receiving itdo? Give the forms in which a 
elerk or collector signs his name. 

B 


126 FEDERAL MONEY. 


If the party indebted gives his note, or written promise to pay, in stead 
of cash, it may be mentioned after the words Mecetved Payment, as in 
Bill 4. 

The person indebted may have paid something on account, or may 
have charges against the party rendering the bill. Such amounts are 
called credits. They are placed below the items of the bill, and are de- 
noted by the letters Cv., as in Bill 5. The balance is obtained by finding 
the difference between the sum of the credits and the sum of the debits. 


EXAMPLES FOR PRACTICE, 


Copy each bill, learn the forms, find the cost of each item, 
insert it in its proper place on the right, add the several amounts, 
and see whether their sum agrees with the given answer :— 


1 
Y N. ¥., Feb. 23, 1865. 
Mr. Henry Rog, 
To Terry & Brown, Dr. 

To 75 yards carpeting, @ $2.50 . . . $ 
‘“* 42 yards drugget, @ $1.874 . 
‘* 6 mats, @ $8.25 . : 
“* 18 rugs, @ $22.30 ; 
‘6 §1 yards oil-cloth, @ $1.10 


$776.25 
Received Payment, 


(2) 
Philadelphia, March 1, 1865. 
Mrs. H. S. SKINNER, 
B’t of R. J. James. 

8 yards linen, @ $1.25. . . . . $ 
12 pair hose, @ .75. ° 
Apair gloves, @ ~~ .95 

6.skeins'silk, -@ .05. ... «+ 

1 piece muslin, 44 yards, @ 55. . 


$41.05 
Received Payment, 
R. J. JAMES. 


If the party indebted gives his note, in stead of cash, where may it be men- 
tioned? What is meant by credits? Where are they placed ? How are they de- 
poted? How is the balance obtained? 


MAKING OUT BILLS. 127 


4 Colunbus, May 1, 1865. 
Messrs. Prums & Nixon, 

To Roserr Surrer, Dr. 

To Y-ink-stands, @ 15;e.% we 2jeos SAS Fh 

‘“‘ 9 boxes steel pens, @ 87a... . 

‘* 8 reams foolscap paper, @ $4. . . 

‘* 5 dozen copy-books, @ 96 ¢. a dozen 

“* 3 rosewood writing-desks, @ $7.50 . 


$68.23 
Received Payment, 
Cates Hunt, 
for Roperr Surrer. 


(4) 
Trenton, April 1, 1865. 


Mr. Benzamin Srarx, 
Bot. of Duptry Starr & Co. 

Jan. 10 8 boxes raisins, @ $6.25 $ 

‘¢ 12 50 pounds sugar, @ 206. 
Feb. 16 48 pounds currants, @ 33} c. 

“* 17 120 pounds tallow, @ 163 c. 
Mar. 23-14 barrels flour, @ $10.75 

‘* 25 380 gallons kerosene, @ 85 c. 


$272.00 
Received Payment, by note, 
Duptry Starr & Co. 


(5) 
Concord, May 10, 1868. 


To Dr. W. 8. Crane, Dr. 
To professional services to date . . . . $68.00 


Mr. Ricuarp Foor, 


se RILCOICHIBS: LO: CALO 0. Gn 16g) Ys. as 4.35 
$72.35 

Dy 9 sok: heal 6 2) unin eee ea pane meme Wf 

‘“ 5 cords wood, @ $6 30. 
— $45.00 
Balaneé 209) @ $27.35 

Received Payment, 
W. 5S. Cranz, 


by Asa Green, 


128 FEDERAL MONEY. 


6. John Cox bought of Philip Brady, of Boston, the following 
articles :—Jan. 2, 1865, 6 pair of gloves, at $1.25; Feb. 1, 12 
shirts, at $3.75; Feb. 9, 18 pair socks, at 834 ¢c.; Feb. 13, 1 over- 
coat, at $40; Feb. 20, 2 vests, at $8.75, and 2 umbrellas, at $3.30. 
Make out Cox’s bill, and receipt it. Ans. $122.60. 

7. James Ray, of Detroit, sold George Mott the following 
articles :—25 pounds beef, at 19 c.; 8 pair fowls, 40 pounds, at 
i21 c.; 50 pounds sausage-meat, at 124 c.; 25 bushels potatoes, at 
50 c.; 45 pounds of lard, at 163c. Mott paid on account $15. 
Make out his bill, and find the balance due. Ans. $24.40. 

8, H. 8. Fair, of Hartford, sold N. T. Wright, 4960 red oak 
hhd. staves, at $90 per M.; 3575 feet hemlock boards, at $26 per 
M.; 2240 feet white oak plank, at $55 per M.; 4785 feet pine 
boards, at $28 per M. The same party bought of N. T. Wright, 
87 yards carpeting, at $1.50; 244 yards matting, at 90 c.; 404 
yards oil-cloth, at $1.40. Make out Fair’s bill against Wright, 
showing the balance due. Ans. $662.63. 

9, William Haight bought of Hiram See, of N. O., 42 boxes 
of oranges, at $8.12; 7640 pounds coffee, at 834 c.; 2400 gallons 
molasses, at 92 c.; 875 pounds rice, at 12 c.; 1250 pounds sugar, 
at 124c. See credits Haight with 400 barrels of flour, at $9.75, 
and takes Haight’s note to balance account. Mako out bill. - 

Ans. Balance, $1456.96, 

10. Mrs. Stewart, of Newark, presents her bill to P. S. How- 
ard for board, &c., as follows :—6 weeks’ board, at $8.25 a week ; 
fuel 6 weeks, at $1.20; gas 6 weeks, at 50 c.; washing, 7 dozen 
pieces, at $1 a dozen. Make out the bill. Ans. $66.70. 

11. Suppose you buy of D. Appleton & Co. 5 reams of note 
paper, at $3.25; 4500 envelopes, at $4.75 a thousand; 24 boxes 
steel pens, at $1.124; 6 French Dictionaries, at $1.50; 8 Photo- 
graphic Albums, at $5.75. Make out your bill. Ans. $90.88. 
' 12. M. Stagg, of Baltimore, sold James Quinn the following 
articles :—April 1, 1865, 24 yd. black silk, at $2.25; April 3, 2 
pieces French calico, 40 yd. each, at 830c¢.; May 2, 4 dress patterns, 
at $6.75; May 9, 224 yd. linen, at $1.12. Quinn paid $55 on ac- 
count. Make out his bill, showing balance due. Ans. $75.20, 


REDUCTION, 129 


MIscELLANEOUS QuEsTIons oN DecimALs AND FeperaL Monry.—What 
does the word decimal come from? How do decimals arise? How can 
you find the denominator of a decimal? Does annexing a naught to a 
decimal increase or diminish its value? When adding decimals, where 
do you place the decimal point in the result? Whensubtracting? When 
multiplying? When dividing? In division of decimals, when will the 
quotient bea whole number? How do we multiply a decimal by 10, 100, 
1000, &c.? How do we divide a decimal by 10, 100, 1000, &e. ? 

What is a circulating decimal? What is arepetend? How do you 
reduce a decimal to a common fraction? How do you reduce a repetend 
to a common fraction? How do you reduce a common fraction to a 
decimal? 

What 1s federal money? How do you write federal money? Why 
are two places appropriated to cents ? How do you read federal money ? 
How do you add, subtract, multiply, and divide federal money? What is 
a bill? When are credits to be entered in a bill? When must a bill be 
receipted? What are the forms to be used by a clerk in receipting a bill? 


CHAPTER XIII. 
REDUCTION. 


216. How many cents in five dollars ? 


100 cents make $1; in $5, therefore, there are 5 times 100 cents, or 
500 cents. Ans. 500 centa, ; 

We have here changed the denomination from dollars to cents, with- 
out changing the value. This process is called Reduction. We have 
reduced dollars to cents. 

217. Reduction is the process of changing the denomi- 


nation of a number without changing its value. 

218. There are two kinds of Reduction :— 

1. Reduction Descending, in which we change a higher 
denomination to a lower, as dollars to cents. Here we 
must multiply. 


216. How many cents in $5? What have we here done? What is this process 
called?—217, What is Reduction ?—218. How many kinds of Reduction are there? 
What are they called? What is Retluction Descending ? 


130 REDUCTION. 


2. Reduction Ascending, in which we change a lower 
denomination to a higher, as cents to dollars, Here we 
must divide. | 


219. Reduction Descending. 


EXAMPLE 1.—Reduce $41 to mills. tt 
100 cents make $1; in $41, therefore, there are eh 
100 times 41 cents, or 4100 cents. 4100 ¢. 

10 mills make 1 cent; in 4100 cents, therefore, 10 


there are 10 times 4100 mills, or 41000 mills. Ans, 41000 m. 


Example 2.—Reduce $41.375 to mills. 


Reduce $41 to cents: 41 x-100 == 410G-¢. 

Add in 87 cents: 4100 + 37 = 41387 c¢. 
Reduce 4137 cents to mills: e137 x 10 => 4101010: 

Add in 5 mills: 41370 + 5 = 41875 m. Ans. 


220. Rute ror Repvuction Derscenpinc.—Multiply 
the highest given denomination by the number that it takes 
of the next lower to make one of this higher, and add in 
the number belonging to such lower denomination, if any 
be given. Goon thus with each denomination in turn, 
till the one required is reached. 


221, Reduction Ascending. 


EXxAMpie 3.—Reduce 41375 mills to dollars. 


10 mills make 1 cent; therefore in 41375 10) 41375 m. 

mills there are as many cents as 10 is contained o____ 

times in 41375, or 4137 cents, and 5 mills over. 100) 4137¢., 5m 
100 cents make 1 dollar; therefore in 4137 

cents there are as many dollars as 100 is con- Ans. $41.375 

tained times in 4137, or $41, and 37 cents over. 

The last quotient and the two remainders form the answer—$41, 37 cents, 

5 mills, or $41.375. 


222. Rute ror Repuction Ascenpinc.—Divide the 
given denomination by the number that it takes of it to 


What is Reduction Ascending ?—219. Solve the given examples, explaining the 
several steps.—220. What is the rule for Reduction Descending ?—221. Reduce 4137 
mills to dollars.—222. What is the rule for Reduction Ascending ? é 


REDUCTION OF FEDERAL MONEY. 131 


make one of the next higher. Divide the quotient in the 
same way, and go on thus till the required denomination 
zs reached. The last quotient and the several remainders 
Sorm the answer, 


223, In Example 2 we reduced $41.375, and obtained 41375 mills. 
In Example 3, we reduced 41875 mills, and obtained $41.875. Thus it 
will be seen that Reduction Descending and Reduction Ascending prove: 
each other. 


224, Reduction of Federal Money. 


In Example 1, § 219, we reduced dollars to cents by annexing two 
naughts, cents to mills by annexing one naught. 

In Example 2, § 219, comparing the result, 41375 mills, with $41.375, 
the amount to be reduced, we find it is the same, with the dollar-mark and 
decimal point omitted. 

In Example 3, § 221, comparing the result, $41.375, with 41375 mills, 
the amount to be reduced, we find that we have simply pointed off three 
figures from the right, and inserted the dollar-mark. Hence the following 
rules :— 


RvLes ror tHE Repvuctrion or Frprran Monry.— 
1. To reduce dollars to mills, annex three naughts ; to 
reduce dollars to cents, two ; to reduce cents to mills, one. 

2. To reduce dollars and cents to cents, or dollars, cents, 
and mills, to mills, simply remove the dollar-mark and 
the decimal point. 

3. To reduce mills to dollars, point off three figures 
Srom the right ; to reduce cents to dollars, two ; to reduce 
mills to cents, one. 


EXAMPLES FOR PRACTICE. 


Reduce the following :— 


1. $68.47 to cents. 6. $.059 to mills. 

9. $5.485 to mills. 7. $2.85 to cents. 

8. $2480 to mills. 8. $5000 to mills. 

4, $56.90 to mills. 9. 2468 mills to cents. 
5. $4283 to cents. 10. 2570 mills to dollars. 


223. How is Reduction Descending proved? Reduction Ascending ?—224. Re- 
cite the rules for the reduction of federal money. 


182 REDUCTION. 
11. 8620 cents to dollars. 14. 56000 cents to mills. 
12. 490000 mills to cents. 15. 8705 cents to dollars. 
18. 56000 cents to dollars. 16. $87.05 to mills. 


17. How many centsin4 eagles? (4eagles = $40) Ans. 4000c. 

18. How many cents is a double eagle worth? Ans. 2000c. 

19. How many eagles are 8000 cents worth? 2000 cents? 

20. Reduce 423756890 mills to dollars. 

21. How many cents in $894? In $1024? In $44? 

22. How many mills in 374 cents? In $5.624? 

23. How many quarter-dollars equal a double eagle? 

94, How many dimes in $1? In $15? In $30? In $49? 

25. How many cents in 1 dime? In 5 dimes? In 20 dimes? 

26. How many dimes are equal to 10 cents? Te 150 cents? 

27. How many half-dollars ought I to receive in change for 
an eagle? For two double eagles? 

28. How many cents is a quarter-eagle worth? A half-eagle? 
A three-dollar piece? A half-dollar? Five dimes? 

29. Reduce each of the following to cents, and add the results: 
2 eagles; 5 half-dollars; 15 dollars; 1 double eagle; 3 quarter- 
dollars; 12 dimes; 120 mills. Ans. 5957 cents. 


Compound Numbers. 


225. A Compound Number is one consisting of different 
denominations; as, 3 dollars 19 cents. 

226. Compound numbers may be reduced, added, sub- 
tracted, multiplied, and divided. 

227. To show the relations that different denomina- 
tions bear to each other, Tables are constructed. These 
are now presented in turn, with examples in Reduction 
under each; they should be thoroughly committed to 
memory. or convenience of reference, these Tables are 
reproduced together on the last page of the book. 


225, What is a Compound Number ?—226. What operations may be performed 
on Compound Numbers ?—227. For what purpose haye Tables been constructed in 
connection with Compound Numbers ? 


>: 
o 
- 
. 
¢ 
af 


STERLING MONEY. 133. 


ENGLISH OR STERLING MONEY. 
228, English or Sterling Money is the currency of 
Great Britain. 
TABLE. 


4. farthings (far.,qr.); 1 penny, . /. d. 


TL27 pence, ASUS, Gu onc S, 
20 shillings, We DOUM cease. eh 
21 shillings, POOUINCA see ewe. peut 
d. far. 
Ss. 1 — A 
£ begat re f= 48 
guin. a = 20 = 240 — 960 
Laie oe De ae 2 ere L008 


The pound mark £ is a capital /, standing for the Latin word libra, a 
pound; it always precedes the number, as £2.. S. stands for the Latin 
solidus, a shilling; d. for denarius, a penny; qr. for guadrans, a farthing. 

Shillings are pneu written at the left of an inclined line, and 
pence at the Weitr -/)se oar | pt COs (ah! 28; 60. Farthings are 
sometimes written as the fraction of a penny, 1 far. as ad., 2 far, as od, 
3 far. as 3d. 

The pound is simply a denomination ; a gold coin called the Sovereign 
represents it. The Sovereign is worth $4.84. The English shilling is 
worth 24+ cents, and the English penny about 2 cents. 

Guineas, originally made of gold brought from Guinea, are no longer 
coined. The Crown isa silver coin, worth 5 shillings. 


229, In the twelfth century, some traders from the Baltic coasts, 
called by the people Easterlings because coming from regions farther east, 
were employed to regulate the coinage of England. From these Hasterlings 
the currency took the name of Sterling Money. — , 


HXAMPLES FOR -PRACTICE. 


230. Recite the rules for Reduction, § 220, 222. 
EXAMPLE 1.—Reduce £5 19s. 3 far. to farthings., 


228. What is English or Sterling Money? Recite the Table of Sterling Moncy. 
What is the pound mark, and where does it stand? What do s., @., and g7., stand 
for? How are shillings sometimes written? Howare farthings sometimes written? 
Is the pound adenomination oracoin? Whatrepresents it? Whatis the sovereign 
worth? The English shilling? The English penny? Why were guineas so called? 
What is the Crown ?—229. From whom did sterling money receive its name ?—230, 
Go through and explain the given examples in Reduction. 


134 


This is a case of Reduction Descending. Multi- 


REDUCTION. 


£5 19s. 8 far. 


ply the £5 by 20, to reduce them to shillings, be- 
cause 20 shillings make a pound. Add in the 19 
shillings. 

Multiply 119s., thus obtained, by 12, to reduce 
them to pence, because 12 pence make a shilling. 
There are no pence in the given number to add in. 

Multiply the 1428d., thus obtained, by 4, to re- 
duce them to farthings, because 4 farthings make a 
penny. Add in the 8 farthings. -dns. 5715 far. 


. 20 


119s. 
12 


1428d. 
4 


5715 far. Ans. 


EXAMPLE 2.—Reduce 15383 far. to pounds, shillings, &c. 


This is a case of Reduction Ascending. 
15383 far. by 4, to reduce them to pence, because 


4) 153838 far. 


4 farthings make a penny. 


Divide 


12) 3845d. 3 far. 

2/0) 32/08, 5d. 
ae 

Ans. £16 53d, 


Divide the quotient, 3845d., by 12, to reduce it 
to shillings, because 12 pence make a shilling. 

Divide the quotient, 320s., by 20, to reduce it to 
pounds, because 20 shillings make a pound. The 
last quotient and the several remainders form the 
answer.—Always mark the denominations through- 


out, as in these examples. 
Examprir 8.—Reduce £457 to farthings. 


We may here proceed as above, or £ABT 
we may somewhat shorten the opera- 20 
tion. Looking under the Table on 


page 133, we find £1 = 960 far. . Then 9140s, 
in £457 there are 960 times 457 far- 12 
things. When, then, the number to 1 99680d. 
be reduced has but one denomination, 4 

we may multiply at once by the num- ~~ 

ber that connects it with the denomi. 438720 far. 


nation required. 
4, Reduce £7 5s. 10d. 3 far. to farthings. 
5. Reduce £47 5s. 2d. 1 far. to farthings. 
6. Reduce £1 514d. to farthings. 
7. In 18s. 8 far. how many farthings? 
8. Reduce 4963 far. to pounds, &c. 
9. Reduce 1°/, to farthings. 


. 10. In £8000 how many pence? 


11. How many farthings in #/,? In 14/-? 


£1 = 96 far. 
£457 
960 


27420 
A113 


438720 far. 


Ans. 7008 far. 
Ans. 45369 far. 
Ans. 981 far. 
Ans. 867 far. 


Ans; £5, 38. 4d.98 far. 
Reduce °/, to pence. 
Prove by reducing the answers obtained back to shillings. 


In 8s. 34d. ? 


12. How many sovereigns are 12480.pennies worth? 
13. How many pence are 840 sovereigns worth ? 


14, Reduce 560 guineas to farthings, 


Ans. 564480 far. 


TROY WEIGHT. 1357 


15. Reduce 118567 far.-to pounds, &e. Ans, £128 10s. 14d. _ 


16. Reduce £3 10s. to pence. Reduce 18s. 9d. to pence. 

17. How many pounds, &c., in 15199 pence? In 189s, ? 

18. How many crowns are £25 equal to? 

1 crown = 5s. How many crowns in £1, or 20s.? How many in £25? 

19. How many pounds are 100 guineas equal to ? 

20. Reduce 7648s. to pounds; to guineas. 

21. Reduce £1000 to farthings. 

22. Reduce 4800000 far. to pounds, &e. 

23. In 24000 far. how many crowns? Ans. 100 crowns. 


24. A subscribes £500 for the poor; B, 500 guineas. Which 
subscribes the most, and how much ? Ans. B £25. 


TROY WEIGHT. 


231, To express weight, three different scales are used, 
called Troy, Apothecaries’, and Avoirdupois Weight. 

232. Troy Weight is used in weighing gold, silver, 
coins, and precious stones; also in philosophical experi- 
ments. 


TABLE. 
24 grains (gr.) make 1 pennyweight,. . pwt. 
20 pennyweights, Pountey2 275, Se 2 6x 
12 ounces, Lepound 26 FOC) Sib; 
pwt. gr. 
Oz. 1 ante 24 
Ib. aoe ee a bea 
Me a ee on ee OT OU 


The Troy pound is the standard unit of weight of the United States 
and Great Britain. It is equal to the weight of 22.794377 cubic inches 
of distilled water, at its greatest density. 


233. The denominations grain and pennyweight take their name from 
the fact that silver pennies were once coined, required by law to equal in 
weight 32 grains of wheat from the middle of the ear, well dried. The 
value of the penny being afterwards reduced, the number of grains in the 


231. Name the different scales used to express weight.—232. For what is Troy _ 


Weight used? Recite the Table of Troy Weight. What is the standard unit of 
weight of the United States? To what is the Troy pound equal ?—233, Why are 
the grain and pennyweight so called ? 


at 


136 


REDUCTION. 


pennyweight was also reduced to 24.—Oz. stands for the Spanish word 


onzd, an OunCE. 


234, Troy Weight takes its name from Troyes, a town of France, 


whence it was carried to England by goldsmiths ; 


or, according to others, 


~ from Troy Novant, an old name applied to London. 


EXAMPLES FOR PRACTION. 


1. Reduce 801b. 3 0z. to pwt. 


Which kind of Reduction does this 
fall under? Recite the rule (§ 220).— 
Multiply the 801b. by 12, to reduce them 
to ounces; addin the 380z. Multiply the 
ounces thus obtained by 20, to reduce 
them to pwt.; there being no pwt. to 
add in, this result is the answer. 


80 1b. 8 oz. 
12 


363 oz. 
20 
~ Ans. 7260 pwt. 


8. Reduce 61b. 40z. 3pwt. 5 gr. to grains. 

4, How many grains in 11 0z. 19 pwt. 23 gr.? 
5. In 120031b. how many pennyweights ? 
6. Reduce 9999 gr. to pounds. 

7. Reduce 999 pwt. to pounds, &e. 
8. Reduce 1561 oz. to pounds, &c. 
9. Reduce 18 pwt. 4 gr. to grains. 

- Reduce 11000 grains to lb. Ans. 1 Ib. 10 0z. 18 pwt. 8 gr. 
. In 25]b. 17 pwt. how many grains? 

. In 87]b. 50z. how many pennyweights? 

. In 8548 grains, how many pounds, &.? 

Reduce each of the following to pounds, and add the re- 
; 6960 pwt. 
15. Reduce the iioasee to grains, and add the results: 


14. 


sults: 40320 er.; 960 oz. 


tpwt19'er. :.11 0z.1 hpwt,; 


2. Reduce 7681 pwt. to pounds, 
&e. 


Which kind of Reduction does this 
fallunder? Recite the rule (§ 222)—As 
20 pwt. make an ounce, divide the given 
pennyweights by 20, to reduce them to 
ounces. Divide the ounces thus obtained 
by 12, to reduce them to pounds. The 
last quotient and the remainder form the 
answer. - 


2/0) '768]1 pwt. 
12) 384 0z. 1 pwt. 
732 Ib. 
Ans. 32\b. 1 pwt. 
Ans. 36557 gr. 
Ans. 5759 gr. 
Ans. 288180 pwt. 
Ans. 1lb. 80z, 16 pwt. 15 gr. 
Ans. 4Ib. loz. 19 pwt. 
Ans. 1301b. 1 oz. 
Ans. 486 gr. 


Ans. 116 Ib. 
5 Ib. 


101b. 40z. ll pwt. Ans. 94314 gr. 


16. How many ounces in four lumps of gold, weighing 7 pwt., 


13 pwt., 15 pwt., and 18 pwt. ? 


Ans, 202. 18 pwt. 


234, From what does Troy Weight take its name? 


APOTHECARIES’ WEIGHT, — 137 


17. What is the weight in pounds of a silver tea-pot weighing 
200 pwt., and 24 table-spoons of 35 pwt. each? Ans. 4]b.. 40z. 
18. How many pounds of gold will a miner dig in a year of 
365 days, if he averages 6 pwt. daily? Ans. 9b. loz. 10 pwt. 


APOTHECARIES’ WEIGHT. 


235, Apothecaries’ Weight is used by apothecaries in 


mixing medicines. They buy and sell their drugs, in 


quantities, by Avoirdupois Weight. 


TABLE. 

20 grains (gr.) make 1 seruple,. . sc. or 9. 
3 scruples, l-dramjctren drs Or e32 
8 drams, POUNCE 47, OZ8-OF 73... 
12 ounces, lepounds 12 1b. For Th: 

8c. er. 

dr. a 20 

OZ. 1 = St 60 

Ib. IgGs ee oe 24 | 480 

fois) IDR Sa OG s 4 288. 575760 


The only difference between Apothecaries’ and Troy Weight lies in the 
Civision of the ounce. The grain, ounce, and pound, are the same in both. 


EXAMPLES FOR PRACTICE. 


1. Reduce 9247 gr. to pounds, &c. Ans. 1b. 7% 23 Ver. 

Which kind of Reduction does this example fall under? Recite the rule (§ 222), 
Name the numbers in order, by which we must divide. Prove the answer by reduc- 
ing it back to grains. 

2. Reduce 9% 63 15 gr. to grains. Ans, 4707 gr. 

Which kind of Reduction does this example fall under? Recite the rule (§ 220). 
Name the numbers in order, by which we must multiply. Why do we not first 
multiply by 12? If drams had been the highest denomination given, by what would 
we have multiplied first? How can you prove the answer ? 


3. Reduce 15648 gr. to pounds, &c. Ans. 2tb.8% 43 2D 8er. 
4, Reduce 1tb. 11% 23 5er. to grains. Ans. 11165 gr. 
5. Reduce 47635 to pounds, &c. Ans. 16 1b. 6 oz. 3 dr. 2:se. 


’ 


235. By whom is Apothecaries’ Weight used? By what do they buy and sell 
their drugs in quantities? Recite the Table of Apothecaries’ Weight. What is the 
enly difference between Apothecaries’ and Troy Weight? 


138 REDUCTION. 


6. Reduce 91b. 5 dr. to scruples. Ans. 2607 se. 
7, Reduce 848 dr. to pounds, &e. Ans. 8ib. 9% 83. 
8. Reduce 141b. 80z. to drams, 

9, Reduce 30019 gr. to pounds, &e. 

10. In 2b. 3% 43 15 11 gr. how many grains? 

11. Reduce the following to grains, and add the results: 


Piped gr. 10242 es Og: Lets lek ey. Ans. 28202 or. 
12. Reduce the following to pounds, and add the results: 
11520 gr.; 960 dr.; 864 sc.; 14402. Ans. 2'7 Vb. 


18. How many doses of 15 gr. each will 5dr. of calomel make ? 

Reduce 5dr. to grains. How many times are 15 grains contained therein ? 

14. How many grains in this mixture: benzoin, 2 3; cascarilla, 
23; nitre, 143; myrrh, 25; charcoal, 3 3? Ans. 2650 gr. 

Reduce each item to grains; then add. 

15. A druggist put up 24 powders of calomel, of 10 gr. each; 
if he had 1 oz. of calomel at first, how many grains will he have 
left? 


AVOIRDUPOIS WHIGHT. 


236. Avoirdupois Weight is used for weighing all arti- 
cles not named under Troy and Apothecaries’ Weight ; 
such as groceries, meat, coal, cotton, all the metals except 
gold and silver, and drugs when sold in quantities. 


TABLe. 
16 drams (dr.) make 1 ounce,. . . . oz, 
16 ounces, T-pOund, a aa8: act 
25 pounds, Liquarter; 3° ¢.9-Qr, 
4 quarters, 1 hundred-wveight, cwt. 
20 hundred-weight, Jton, . . he 
OZ. dr. 
Ib. Los 16 
qr. : 16 ee 256 
ewt. i pe ys aera 400 = 6400 
1 bese) ANPS O00 NES 01 6001 99829600 
Tee BOS 2000 = a A000 ee oer. A000 


236. For what is Avoirdupois Weight used? Recite the Table. 


AVOIRDUPOIS WEIGHT. 139 


237. Avoirdupois is derived from the French words avoir, property, 
and poids, weight.—Cwt., the abbreviation for hundred-weight, is formed 
of ¢c for centum, one hundred, and wé for weight. 


238. Formerly 28 pounds made a quarter, and 112 pounds a hundred- 
weight, in the United States, as they still doin Great Britain. But it is 
no longer customary to allow 112 pounds to the hundred-weight, except in 
the ease of coal at the mines, iron and plaster bought.in large quantities, 
and English goods passing through the Custom House. 

Twenty hundred-weight of 112 pounds make a ton of 2240 pounds, 
which is distinguished as a Long or Gross Ton. 


239, The Avoirdupois pound weighs 7000 grains Troy, and is there- 
fore greater than the Troy pound, which contains 5760 grains. The 
Avoirdupois ounce weighs 4374 grains, and is therefore less than the Troy 
ounce, which contains 480. grains. : 

1db, Avoir. = 7000 gr. = 1b. 202, 11 pwt. 16 gr. Troy. 
1 oz. Avoir,. = 4374 gr. = 18 pwt. 5} gr. Troy. 

1 1b. Troy or Apoth. = 5760 gr. = 138%; oz. Avoir. 

1 oz. Troy or Apoth. = 480gr. = 1/5 oz. Avoir. 


EXAMPLES FOR. PRACTIOE. 


1. Reduce 10 cwt. to drams. 

_ Looking among the equivalents under the Table, we find lewt. = 25600 dr. 
Then 10 cwt. = 10 x 25600 dr. -Azs. 256000 dr.—When there are no intermediate de- 
nominations, the Table of equivalents can thus be used with advantage. 

2. Reduce 4815 1b. to hundred-weight. 

In the Table of equivalents we find 1ewt. = 1001lb. Then in 4815 1b. there are 
as many ewt. as 1001b. are contained times in 48151b. Ans. 48 cwt. 15 1b. 

3. Reduce 3-T. 15 cwt. 16]b. 5 oz. 5dr. to drams, 

Which kind of Reduction does this example fallunder? Repeat the Rulc (§ 220), 
What numbers must we multiply by? Prove the result. Ans, 1924181 dr. 

4, Reduce 294400 oz. to tons, &. 

Which kind of Reduction does this example fall under? Repeat the Rule (§ 222). 
Mention the successive divisors, Prove the result. 

5. Reduce-1 T. 15 Ib. to ounces, Ans. 32240 oz. 

6. Reduce 1792512 dr. to tons, &c. Ans. 8T. 10 cwt. 2Ib. 


7. How many pounds in two loads of 24 tons each ? 


237. From what is the word avoirdwpois derived? Of what is the abbreviation 
ewt. formed ?—238. How many pounds formerly made a hundred-weight? In what 
alone is it now customary to allow 112 1b. to the hundred-weight? What is a Long 
Ton ?—239. How many grains in the avoirdupois and the Troy pound respectively ? 
In the ayoirdupois and the Troy ounce? What is 11b. avoir. equivalent to in Troy 
weight? What is 11b. Troy equivalent to in avoirdupois weight? 


140 REDUCTION. 


8. How many pounds in four loads of 84 tons each ? 

9. How many drams in 123 tons? 

10. Reduce 24 1b. 3 oz. 14dr. to drams. 

11. How many tons, &c., in 94500 oz. ? 

12. Reduce 2T. 2 cwt. 2 qrij2Ib. 20z. 2dr. to drams. 


13. How many drams in 27 long tons ? Ans. 15482880 dr. 


14, In 424 long tons how many pounds? Ans. 95200 Ib. 
15. Reduce 5T. lewt. 13]b. to drams. Ans, 2588928 dr. 
- 16. Reduce the following to drams, and add the results: 74 
tons; 24 long tons; llcwt.; 41b. 402. Ans. 5284928 dr. 
17. Reduce the following to hundred-weight, and add the re- 
sults: 6400 0z.; 1700 Ib.; 281600dr.; 28 qr. ~ Ans. 39 cwt. 


18. How many more pounds in 1 long ton than 1 common 
ton? In 25 long tons than 25 common tons? 

19. Ifa coal-merchant buys a cargo of 200 long tons, and sells 
200 common tons, how many pounds has he left? How many 
common tons? How many long tons? Ans. 21# long tons. 

20. How many two-ounce weights can be made out of 50 
pounds of brass ? 

Tiow many oz. in 501b.? How many times are 2 oz. contained therein ? 

21. How many five-pound weights can be made out of 54 cwt. 
of iron? Out of 64 cwt. ? 

22. How many more grains in 1b. avoirdupois than in 1 1b. 
Troy? (See § 239.) In 14]b. avoir. than in 14 Ib. Troy? 

~ 28, How many pounds Troy are 144 Ib. avoir. equal to ? 


How many grains in 1]b. avoir.? How many in 144 Ib. avoir.? How many Ib. 
Troy in these, if 11b. Troy contains 5760 grains? 


24. Reduce 1225 Ib. Troy to avoirdupois pounds. 


25. Reduce 875 oz. apothecaries’ weight to pounds avoir. 


How many grains in 1 oz. apoth.? How many in 8750z.? Reduce these grains 
to pounds avoirdupois. 


26. Reduce 2880 oz. avoir. to Troy ounces. Ans. 2625 02. 
27. What cost 544 cwt. of pork, at 1lc.a pound? Ans. $599.50. 
28. What cost 26 cwt. of hams, at 6d. a Ib. ? Ans. £65. 
29. What cost 9 T. of iron at 13d. a Ib. Ans. £131 5a, 


30. What cost 475 T. of iron, at $4.50 a cwt. ? 
81. What cost 100% cwt. of cheese, at 10 c¢. a pound? 


we ; 9 Seto xX 
"MISCELLANEOUS TABLE. — eee 
\ q a 


240. Miscieiate Pinin “Ky / 


The pounds in this Table are avoirdupois. 


14 pounds, . . 1 stone of iron or lead. 

60 pounds, . . 1 bushel of wheat. 
100 pounds, . . 1 quintal of dried fish. 
100 pounds, . . 1 cask of raisins. * 
196 pounds, . . 1 barrel of flour. 
200 pounds, . . 1 bar. of beef, pork, or fish. oS 
280 Bongs: . . 1 bar. of salt at the N. Y. State works, * 


EXAMPLES FOR. PRACTIOE. 


1. How many ounces in 14 stone? . Ans. 8186 02. 

2. How many stone are 7 cwt. equal to? _ Ans. 50st. 

Reduce Tewt. to pounds. Divide by the number of pounds in 1 stone. 

3. How many barrels will 98 cwt. of flour make? 

4, At 7c. a pound, what cost 46 quintals of cod-fish ? 

5. How many bushels in 330 lb. of wheat? 

6. If flour is $9.80 a barrel, how much is that a pound ? i. oe 

7. How many hundred-weight in 25 barrels of salt bought - : . ; ns 

at the N. Y. State salt works ? Me) 

8. How many seven-pound boxes can be filled from 21 casks 
of raisins? | 

9. If $ of a barrel of flour is sold, how many pounds remain 
in the barrel ? 

10. At 2d. a pound, what cost 125 quintals of dried fish ? 


LONG OR LINEAR MEASURE. 


241. There are three dimensions: length, or distance 
from end to end; breadth, or distance from side to ae 4 
and thickness, o distance fe om top to bottom. : 

A line has length; a surface, length and breadth ; a8 
solid, length, breadth, and thickness, = 

240. Recite the Miscellaneous Table. What kind of Seah ace these ?—241. Tow 


many dimensions are there? Name and define them, Which of these dimensions 
hasaline? A surface? <A solid? 


142 . REDUCTION, 


242, Long or Linear Measure is used in measuring 
length and distance. It begins with the inch. 


1 inch. 
TABLE, 
@ 12 inches (in.).make 1 foot, . ..... ft. 
3 feet, Layard, .. abaneayer 
54 yards, 1 tod, p< Si tment ae 
40 rods, 1 funtony, fats 
8 furlongs, Limile, iy Sete 
ft in. 
yd. pe 12 
rd. = — Dy tes 36 
fur. Loe a 163 = 198 
m. Pe ey eh ee, OD a G0 a meee 
l= 6 = 8000) =. 1700 —* OaeU = ego 0U 
243, The following denominations also occur :— 
The Line = + inch. The Pace = 8 feet. 
The Hand = 4 inches. The Fathom = 6 feet. 
The Span = 9 inches. The Geographical Mile = 1345 + mi. 
The Cubit = 18 inches. The League = 3 miles, 


The Hand is used in measuring the height of horses; the Fathom, in 
measuring depths at sea. The mile of the Table (5280 feet) is the land 
mile recognized by law in the United States and England, and is therefore 
distinguished as the Statute Mile. The land league consists of 3 statute 
miles; the nautical league, of 3 geographical or nautical miles.—A vessel 
is said to run as many ‘nots as she sails geographical miles in an hour.—. 
Rods are sometimes called poles, or perches. 


244, Crora Mrasure.—In measuring drygoods, as 
cloth, muslin, &c., the yard of long measure is used, divid- 
ed into halves, quarters, eighths, and sixteenths. 'The six- 
teenth of a yard, also called a Nail, contains 21 inches. 


242. For what is Long or Linear Measure used? Recite the Table of Long 
Measure.-—243. To what is the Line equal? The Hand? The Span? The Cubit? 
The Pace? The Fathom? The Geographical Mile? The League? What is tho 
Hand used in measuring? The Fathom? THow is the mile of the Table (5280 
feet) distinguished? Of what does the land league consist? 'The nautical league? 
What is meant by 2 vessel’s running a certain number of knots ?—244. What is 
used in measuring drygoods? What is the sixteenth of a yard sometimes called? 
How many inches in a Nail? 


LONG MEASURE. 143 


The Ell, which in Flanders consists of 3 qr., in Eng- 
land of 5qr., and in France of 6 qr., is not used in the 
United States. 


EXAMPLES FOR PRAOTIOLR. 


1. Reduce 3534 inches to rods, &ec. 
Divide by 12, to reduce to feet. Di- 


12) 3534 in. vide the quotient by 8, to reduce to yards. 
oer : Divide the quotient by 54, or 44, to reduce 
3) 294 ft. 6 in. to rods. To divide by ue multiply by 

98 yd. the fraction inverted ;*4;. Multiplying by 

2 2 reduces the yards to half-yards, and on 

7 dividing by 11 we get 17rd., and 9half- 

ee half-yd. yards remainder. But 9 hf. yd = 4h yd. 

Ivrea: ShLsyd.v — 4 yd. 14 ft. Adding to this the first re- 

17rd. 4yd. 14 ft. mainder, 6 inches = }ft., we get Ans. 
arte 17rd. 4yd. 2 ft. 


After multiplying by -7;, therefore, to 
reduce yards to rods, if there is a remain- 
der, divide it by 2, to bring it to yards. 

2. Reduce 5 mi. 3fur. 10rd. to inches. Ans. 342540 in. 


Multiply 5mi. by 8, and add in 8. Multiply this result by 40, and add in 10. 
Multiply by 54, by 8, by 12. 


3. In 7860 inches how many rods, &c.? 
4, In 64 miles how many feet? (5280 x 64) 
5. Reduce 6 fur. 5rd. 1 yd. 2 ft. to inches. 
6. Reduce 12012 inches to rods, &c. Ans. 60rd. 8 yd. 2 ft. 
7. How many inches in 82 miles? Ans. 237600 in. 
8. Reduce 54954 inches to furlongs, &c. Ans. 6fur. 87rd. 8 yd. 
9. Reduce 134507 ft. to miles. Reduce 5000rd. to miles. 
10. How many leagues (§ 243) in 9600 rods? Ans. 10 leagues; 
11. How many feet high is a horse whose height is 15 hands? 
12. How many pacesin1 mile? In10rods? _ 
18. Reduce 14640 ft. tomi. Ans. 2mi. 6fur. 7rd. Lyd. 1ft. 6in. 
14, Reduce 87844 in. to higher denominations. 

Ans. 1mi., 3for. 8rd. 8yd. 1 ft. 10 in. 


Ans. 17rd. 4yd. 2 ft. 


How many quartersinthe Ell of Flanders or Flemish Ell? In the English Ell? 
In the French Ell? Is the ell used in the U. 8.? Solve Example 1, explaining the 
steps. After multiplying by ,3,, to reduce yards to rods, if there is a remainder, 
wat must be dene with it? 


144 REDUCTION. 


15. Reduce the following to miles, and add the results: 


60720 ft.; 12 fur.; 126720in.; 8800 yd. Ans. 20 miles. 
16. Reduce the following to inches, and add the results: 39rd. 
2 {t.: Gfur, 510. (1 mi. 5 yaa tte Ans. 118823 in, 


17. How many times will a wheel 6 ft. around, turn in going 
5 miles? 

How many feet in 5 miles? How many times are 6 ft. contained therein ? 

18. In 108 inches how many cubits? (See $243.) How many 
spans? How many hands? How many lines? 

19. How many feet deep is the water in a certain bay, if 
soundings show a depth of 140 fathoms? - 

20. About how many statute miles are 15 geographical miles 
equal to ? 

21. How long will it take a vessel running 10 knots to sail 12 
nautical leagues ? Ans. 32 hours. 

22. Sound moves 1120 ft. in a second. How far off is a thun-: 
der-cloud, when the clap is heard 11 seconds after the flash is 
seen ? Ans. 24 mi. 

23. How many inches long is a piece of muslin containing 44 
yd.?. How many nails in the same piece? (16 nails = 1 yd.) 

24, Bought three pieces of silk containing 37, 38, and 39 yards. 
How many pieces half a yard long can be cut from them ? 

25. How many sixteenths in 234 yards? 

26. How many nails in 44 yards of cloth? 

27. What cost 5 yd. 1 nail of cloth, at $6.40 a yard? 

1nail = J; yd. $6.40 x 53. 


SURVEYORS MEASURE. 


245, A Surveyor is one who measures land. In meas- 
uring land, Gunter’s Chain (so called after an eminent 
English mathematician, who invented it) is commonly 
used. Its length is 4 rods, or 22 yards, and it is divided 
into 100 links. 


245, What is a Surveyor? In measuring land, what is commonly used? How 
long is Gunter’s Chain ? 


SURVEYORS’ MEASURE. “. P45= 


TABLE. 

7.92 inches (in.) make llink,. . . 21 
100 links, 1 Ghai of 400)" ch, 
80 chains, 1 matey 2. Fer em 

1 in. 
ch. hous 7.92 
mi. Ae ee EL OEE 792 
1 =z 80. = 8000 = 68360 


Links may be written decimally, as hundredths of a chain. 4 ch. 821. 
= 4,32 ch. 


246, 1 chain =4rods. Hence, io reduce chains and links to rods, 
write the links as the decimal of a chai, and multiply by 4. Multiply 
this result by 54 to reduce to yards, or by 164 to reduce to feet. 


EXAMPLES FOR PRAOTIOE. 


1. Reduce 40ch. 25 1. to feet. 

40,25 x 4 = 161 reds = 26563 ft. Ans. 

2. Reduce 3 ch, 151. to inches. Ans. 2494.8 in. 

3. A surveyor finds the distance between two bridges to be 
340 chains; how many miles apart are they ? 

4, A farmer runs a fence on each side of a lane 20 chains in 
length. How many yards of fence does he put up? Ans. 880 yd. 

5. An oblong field is 15 ch. in length and 


10 ch. in width. How many feet long is the 15 ch. 
5 ms 

fence that encloses it ? Ans. 8300 ft. 8 = 
The field has four sides, two of thenf 15 ch. long,and =~ 15 ch. a 


two 10 ch. long. Find, by addition, the length of all four 
sides in chains; then reduce to feet. 


6. How many rods long is a fence that surrounds an oblong 
field 12 chains long and 9 chains wide? Ans. 168 rd. 
7. A man walks round a three-sided field, whose sides measure 
respectively 10, 8, and 4 chains; how many yards does he walk? 


SQUARE MEASURE, 


& 
247. Square Measure is used in measuring surfaces; 
such as land, the walls of rooms, floors, &c. 


Recite the Table of Surveyors’ Measure. How may links be written ?—246, Give 
the rule for reducing chains and links to rods.—247. In what isSquare Measure used? 


7 


Ny 


146 REDUCTION. 


248, A Square is a figure that has A Square Ixcr. 
four equal sides perpendicularone to [{[ i1inn 
another,—that is, leaning no more to 
one side than the other. 


: = . 
A Square Inch is a square whose | é. 
sides are eachaninchlong. ASquare |7 ; 
Foot is a square whose sides are each é 
a foot long. | ame: 
TABLE. 
144 square inches (sq, in.), 1 square foot, sq. ft. 
9 square feet, 1 square yard, sq. yd. 
301 square yards, 1 square rod, sq. rd. 
40 square rods, A by forays Melee rise The 
4 roods, Liacrey 1b. 2 esanh 
640 acres, 1 square mile, sq. mi. 
sq. ft. sq. in. 
sq. yd Wess 144 
Ss sq. rd. a= W) = 1296 
a 9 R. 1= 30b= 2791 = 39204 
ai? Oys'a.4. I= 40= 1210= 10890= 1568160 
Vogm. 1= 4= 160= 4840= 43560= 6272640 


\ WL = 640 = 2560 = 102400 = 3097600 = 27878400 = 4014489600 


249, Stone-cutters often estimate their work by the square foot; plas- 
terers and pavers, by the square yard. 


950, 12 inches make a foot, but 
144 square inches make a square foot. 
Why ?—Look at the figure on the right. Bs 
Suppose each of its sides to be 1 foot 3 
long; it will then represent a square ¢ 
foot. Each side is divided into 12 equal > a 
parts, representing inches. By drawing 
lines across the figure from the inch 
divisions, we form a number of small , 
squares, ‘each of which represents 1 
square inch. It will be seen that the 
1 sq. ft. contains 12 rows of 12 square 
inches each, making in all 144 sq. in. ° 


So, l yd. = 8 ft. Then 1 sq. yd. = 3 x8 (9) sq. ft. 
lrd. = 55 yd. Then 1 sq. rd. = 54 x 54 (804) sq. yd. 


948, What is aSquare? What isa Square Inch? A Square Foot? Recite the 
Table of Square Measure.—249. How do stone-eutters, plasterers, and pavers often 
estimate their work 7-250. Show why itis that 144 square inches make 1 square foot. 


1 foot=12 inches. 


1 foot = 12 in 


I 


i 


SQUARE MEASURE. 147 


251, Roods and acres have no corresponding denomination in linear 
measure ; hence we do not say square roods or square acres.—A square 
rod is also called a pole or perch (P.); and a square mile of land, a section, 
A township is a subdivision of a county, containing 36 square miles or 
sections. 


252. The space contained in a surface is called its 
Area, or Superficial Contents. To find the area of a 
four-sided figure whose sides are perpendicular one to 
another, multiply the length by the breadth. 


The length and breadth must be in the same denomination, and the 
answer will be in the corresponding denomination of square measure. 

Thus, in the figure, the length is 12 in., the breadth 12 in.; the area 
is 12x12 ‘sq. in. A length of 12 in. and breadth of 2, give an area of 
12 x 2 sq. in., as will be seen by counting the squares in the two uppermost 
rows of the figure. A length of 12 in. and breadth of 3, make an area of 
12 x 8, or 36, | sq. in., &e. 

253. Surveyors, taking the dimensions of land in chains, 
on multiplying the length and breadth together, get the 
area in square chains, 10 of which make an acre. Hence, 


to reduce square chains to acres, divide by 10. 


EXAMPLES FOR PRACTIOE. 


1. Reduce 10638 sq. ft. to square rods, &e. 


Divide by 9, to reduce to sq. yds. 9) 10638 sq. ft. 
Divide the quotient by 304, or 124, to 1189 q 
reduce to sq. rods. To divide by. 121, ri sq. yd. 


multiply by the fraction inverted ar 
Multiplying by 4 reduces the sq. yds. to 121) 4728 (89 sq. rd. 
quarters of a sq. yd., and on dividing by 363 

121 we get 39 sq. rods, and 9 quarters 


of asq. yd. remainder. Reduce the re- reee 
mainder to sq. yards by dividing by 4. 
After multiplying by 7$;, therefore, “4/9 9 quarter-sq. -yd. 
to reduce square yards to square rods, “oO sq. yd. 
if there is a remainder, divide it by 4, to 
bring it to square yards. Ans. 39 sq. rd. - 24 sq. yd. 


2. Reduce 1793664 sq. in. to roods. Ans. 1R. 5 P. 22%8q. yd. 


251. Why do we not say square roods or square acres? What is a square rod 
also called? WhatisaSection? What is a Township ?—252, What is meant by the 
Area or Superficial Contents of a surface? Give the rule for finding the area of « four- 
sided figure whose sides are perpendicular one to another. What will be the denomi- 
nation of the answer? Apply this rule in the given example.—253. How many 
square chains make an acre? Give the rule for reducing square chains to acres, 


Ass: 
148 REDUCTION. 
. 8. Reduce 3 A. 27 sq. rd. to square inches. Ans. 19876428 sq. in. 
4, Reduce 1118448 sq. in. tosq. rods. Ans, 28sq.rd. 16 sq. yd. 
5. In 3.sq. mi. how many perches ? 
6. How many acres in 14 sections (§ 251) ? 
7. How many acres in a township (§ 251)? 
8, Reduce 262683 sq. ft. to acres, &c. Ans. 6A. 4P. 26 sq. yd. 
9. Reduce 45 A. 8 R. 21 P. to poles (§ 251). Ans. 7341 P. 


10. How many sq. yards ina garden 5rd. long by 4rd. wide? 

See §252. 5rd. x 4rd. = 20s8q.rd. Reduce 20-sq. rd. to sq. yards. 

11. How many sq. yards in a court, 20 ft. long, 18 ft. wide? 

12. A piece of land is 45 chains in length and 30 in breadth. 
How many acres does it contain ($ 253) ? Ans. 185 A. 

138. How many acres in a field, 40rd. long, 24 rd. wide? Ans. 6. 

14. How many square rods in a garden 100 feet by 90? 

15. In a tract measuring 60 chains in length and 58.50 chains 
in width, how many acres? Ans, 821 A. 

16. How many square yards of oil-cloth will be required to 
cover an office 18 feet by 14 feet ? 

17. How many yards of yard-wide carpeting will be needed to 
cover a room 27 feet by 16 feet? Ans. 48 yd. 

18. At 35 cents a square yard, what will it cost to plaster a 
wall 15 feet high and 54 feet long? Ans. $31.50. 

19. What will be the cost of a piece of land 80 rods square, at 
$45.50 an acre? Ans. $1820. 


CUBIC MEASURE. 


204, Cubic Measure is used in measuring bodies, which 
have length, breadth, and depth or thickness; as stone, 
timber, earth, boxes, &c. 

205, A Cube is a body bounded by six equal squares. 

A Cubic Inch -is a cube, one inch long, one inch broad, 
and one inch thick. Each of its six sides, or faces, is a 
square inch. 


254. In what is Cubic Measure used ?—255, What is a Cube? What is a Cubic 
Inch? 


4 


CUBIC MEASURE. 149 


The engraving represents a Cubic Yard. 
It is 1 yard, or 3 feet, in length, breadth, 
and depth. It will be seen that each of x 
its six faces is 1 square yard, or 9 (3 x3) 
square feet. sé 
The top of this cube contains 9 square 
feet. Hence, if it were only 1 foot deep, ° 


it would contain 9 cubic feet. As it is 3 Bl] 
feet deep, it contains 3 times 9, or 27, a 
cubic feet. Hence 27 cubic feet make 1 | 
cubic yard. 


So, 12x 12x12, or 1728, eubic inches make 1 cubic foot. 


TABLE. 

1728 cubic inches (cu, in.), 1 cubic foot, cu. ft. 
27 cubic feet, 1 cubic yard, cu. yd. 
40 cu. ft. of round, or 4 

1 ton or load, 'T. 
50 cu. ft. of hewn timber, ; 
16 cubic feet, | 1 cord foot, .. cd. ft. 
8 cord feet, Ds COTE Pir anee (aes 
cu. ft. cu. in. 
cu. yd. i= 1728 
ed. ft. 1 ERs) HDT is OE GBG 
Cd. Tes 16 ert 27645 
Ld eG a ¥ VES IANS CS? SQV ISL 


256, The ton in this Table is a measured ton; the avoirdupois ton 
is a ton of weight. Round timber is wood in its natural state. A ton of 
round timber consists of as much as, when hewn, will make 40 cubic feet. 


257. A cord of wood isa pile, 8 ft. long, 4 ft. wide, and 4 ft. high. 
Multiplying these dimensions together, we find 128 cubic feet in the cord. 
One foot in length of such a pile is called a cord foot. 


258, Cubic Measure is used in estimating the amount of work in solid 
masonry, in digging cellars, making embankments, &c. 

259. The space contained in a cube or other solid is 
called its Solidity, or Solid Contents. To find the solid 
contents of a body with six faces perpendicular one to 
another, multiply its length, breadth, and depth together. 


What does the engraving represent? How does it show that 27 cubic feet make 
l1cubic yard? Recite the Table of Cubic Measure.—256. How does the ton in this 
Table differ from the avoirdupois ton? What ismeant by round timber ?—257. What 
is meant by a cord of wood ?—258. What is Cubic Measure often used in estimating ? 
—259. What is meant by Solidity or Solid Contents? Give the rule for finding the 
salidity of a body with six faces perpendicular one to another. 

Cc 


150 REDUCTION. 


The dimensions must be in the same denomination, and the answer 
will be in the corresponding denomination of cubic measure. Thus, let it 
be required to find the solid contents of a box, 6 ft. long, 4 ft. wide, and 
36 inches deep. 


36 in. — 3 ft. 6x4x38 = 472 cu. ft. Ans. 


EXAMPLES FOR PRACTIOE. 


1. How many cubic inches in 484 cu. yd.? Ans. 2029586 cu. in. 

2. Reduce 264884 cu. in. to cu. yd. Ans. 5 cu. yd. 18 cu. ft. 

3. How many cubic feet in 120 cords? 

4, How many cords of wood in a pile, 25 feet long, 4 feet 
wide, and 8 feet high ? 

25x 4x8 = 800 cu. ft. 800 + 128 = 61 Cd. Ans. 

5. How many cords in a pile of wood, 48 feet long, 4 feet 
wide, and 10 feet high ? Ans. 15 Cd. 

6. Reduce 56 cubic yards, 26 cubic feet, 948 cubic inches, to 
cubic inches. 

7. What will it cost to dig a cellar, 30 ft. long, 20 ft. wide, 


and 9 ft. deep, at 624 cents a cubic yard? Ans. $125. 


How many cubic feet in the cellar (§ 259)? How many cubic yards? Multiply 
by the price per cubic yard. 


8. At 75 cents a cubic yard, what will it cost to dig a cellar, 
36 ft. long, 18 ft. wide, and 10 ft. deep ? 

9. What will it cost to make an embankment containing 
999999 cu. ft. of earth, at 70 cents a cubic yard? 

10. At $3.50 a cord, what is the value of a pile of wood, 32 


ft. long, 4 ft. wide, and 7 ft. high ? Ans. $24.50. 
11. At £1 5s. acord, what is the value of a pile of wood, 48 ft. 
long, 103 ft. high, and 4 ft. wide? Ans. £20. 


12. How many cubic inches in 88 cords of wood ? 


LIQUID MEASURE. 


260. Liquid or Wine Measure is used in measuring 
liquids generally ; as, liquors (beer sometimes excepted), 
_ water, oil, milk, &c. 


260. In what is Liquid or Wine Measure used ? 


LIQUID MEASURE. 151 


TABLE. 
4 gills (gi) make Ipmt, .. . . pt. 
2 pints, Liquarty asc verse ae qt 
4 quarts, a Poni eye eee: el, 
314 gallons, Libarrel sgn wet etoasy salts 
2 barrels (63 gal.), 1 hogshead,. . . hhd. 
2 hogsheads, LPC} ie tees PR 
2 pipes, ISSUNss wisn yuaer tl 
pt. gi. 
gal. : Na gems Pfam 8 
bar. Wiens em sui 32 
hhd. bg Ge SLE Hen S126 eae 7 OR Are OUR 
pi. Tees a Oo. tae he ean. a DOA ace 2G 
fun bp 6aloD Fas Ad 2B eee 2604 7 SE erO08 Fa £032 
Daan Pawo oa ho 252s he I OO8 ae 201 GoGo 


42, gallons make 1 tierce (tier.); 2 tierces, 1 puncheon (pun.). 


961. Liquids are put up in casks of different sizes, called barrels, 
tierces, hogsheads, puncheons, pipes or butts, and tuns; but these casks 
seldom contain the exact number of gallons assigned them in the Table. 
The contents are found by gauging, or actual measurement.—When the 
barrel is used in connection with the capacity of cisterns, vats, &c., 31} 
gallons are meant; in Massachusetis, 32 gallons. 


262, The wine gallon of the United States, which is the same as the 
Winchester wine gallon of England, contains 231 cubic inches. The Im- 
penal gallon, established in Great Britain by act of Parliament in 1825, 
contains 277.274 cubic inches, or about 1.2 of our wine gallons. 


EXAMPLES FOR PRAOTIOE. 


1. Reduce 30 gal. 3 qt. 1 pt. to gills. Ans. 988 gi. 


Multiply 30 gal. by 4, to reduce them to quarts, and add in3 qt. Multiply the 
quarts thus obtained by 2, toreduce them to pints,and add ini pt. Multiply the 
pints thus obtained by 4, to reduce them to gills. 


2. Reduce 72 gal. 1 pt. 3 gi. to gills. Ans, 2311 gi. 
3. Reduce 180024 gi. to hhd., &. Ans. 89 hhd. 18 gal. 3 qt. 
4, How many pipes are needed, to hold 23184 pt. of wine? 


Recite the Table. How many gallonsin «tierce? Inu puncheon?—261. Name 
the casks of different sizas in which liquids are put up. How are their contents 
found? When the barrel is used in connection with the capacity of cisterns, how _ 
many gallons are generally meant? How many in Massachusetts ?—262. How many 
subic inches does the wine gallon of the United States contain? Thc Winchester 
wine gallon of Engiand? The Imperial gallon? 


152 , REDUCTION. 


a4 S ‘ 4 

5. How many barrels in 2100 gal. ? 2100 x 2 = 4200 
As many as 31 gall are contained times in i ils 

_ 2100 gall, 81 =%% Multiply by the divisor 4200+ 63 = 66, 42 rem, 


inverted, 2,. Multiplying by 2 reduces the gal- 42-2 = M1 
lons to half-gallons, and on dividing by 63 there 
is a remainder of 42 half-gallons, which we divide Ans. 66 bar. 21 g al. 


by 2, to reduce them to gallons, 
6. How many quarts in 34 hogsheads? 
7. How many pints in 1 tierce, of 42 gallons? 
8. How many gills in 1 hd. holding 61 gall. 8 qt. 1 pt. ? 
9. How many pints in 8 tuns? 
10. What cost 15 gal. of kerosene, at 20c.aqt.? Ans. $12. 
11. What cost 24 qt. of wine, at $5.50 a gal. 2 Ans. $33. 


12. What cost 32 qt. of oil, at 9s. a gal. ? Ans. £3 12s. 
13. How many quart bottles can be filled from a puncheon 
of rum? Ans. 336 bottles. 


14. How many gallons will a cistern hold that has a capacity 
of 10 barrels ? 
15. Reduce the following to gills, and add the results: 15 gal. 


1 pt.; 19 gal. 3qt.; L$ pt. Ang. 1123 gills. 
16. Reduce the following to gallons, and add the results: 
740 qt.; 608 gi.; 312 pt. Ans. 248 gal. 


17. A milkman mixes a gill of water with every pint of milk. 
How many gallons will he thus make out of 48 quarts of pure 
milk? Ans. 15 gal. 


BEER MEASURE. 


263. Beer Measure was formerly employed in measur- 
ing beer and milk. It is now but little used, wine meas 
ure having for the most part taken its place. 


TABLE, 
2 pints (pt.} make 1 quart, . . . . qt. 
4 quarts, Tryallon svete ene 
36 gallons, IL darrelyy ¢: si scqtibar. 


14 barrels (54 gal.), 1 hogshead,. . . hhd. 


263. In what was Beer Measure formerly employed? What is said of its USC as 
the present day? lRecite the Table of Beer Measure. 


DRY MEASURE. ~ 153 yrs 
qt. pt. 
gal. bd — 2 
bar. ie A teens 8 
bha. L >= 86, eee ieee 288 
BP saat hd en OA) ee DL Ge" S482 


The beer gallon contains 282 cubic inches. The gallon, quart, and 
pint of this measure, are therefore greater than those of Wine Measure. 
1 gal. beer measure = 144% gal. wine measure. 


EXAMPLES FOR PRACTICE. 


. Reduce 3} hhd., beer measure, to quarts. 

. How many quarts in 5 barrels, beer measure? 

. Reduce 9640 pt. to barrels, beer measure. 

At 7c. a quart, what cost 5 bar. of beer ? Ans. $50.40. 

. What costs 1hhd. of porter, at 12c.aqt.? Ans. $25.92. 

. Ifa barrel of ale costs $11.52, what is the cost per pt. ? 

7. One third of a hhd. of porter has leaked out. How many ~ 


aor wo NW 


quart bottles can be filled from what remains? Ans. 144. : 
8. If aman buys a barrel of beer for $8.75, and retails it at ; 
9c. a quart, how much does he make ? Ans, $4.21. 


DRY MEASURE. 


264, Dry Measure is used in measuring grain, seeds, 
vegetables, roots, fruit, salt, coal, and other articles not 
liquid. 


TABLE. 
2 pints (pt.) make 1 quart, . ©. . . qt. 
8 quarts, Pr peeks He On pk 
4 pecks, aL, Dustiel CRA ere feb us Par 
36 bushels, Lichaldron;.)%).estarchal 
qt. pt. 
pk. tiie 2 
bu. deo ap 16 
chal bess Ayeics 82a 64 
Dror een BG eck ss . eee D159 se’ 2 2804 


How many cubic inches does the beer gallon contain? How many wine gallons 
does 1 beer gallon equal ?—264. In what is Dry Measure used? Recite the Table. 


154 ~- REDUCTION. 


265, The U.S. standard bushel is the Winchester bushel of Great 
Britain, which contains 2150.42 cubic inches. 

1 qt. of Dry Measure = 1} qt. nearly of Wine Measure.—What is called 
the Small Measure contains 2 quarts. 


266. Foreign coal is imported by the chaldron. American coal is 
bought and sold, in large quantities, by the ton; in small quantities, by 
the bushel. 


EXAMPLES FOR PRAOTIOE. 


1. Reduce 23 bu. 2 pk. 7 qt. to pints. Ans. 1518 pt. 
2. Reduce 18564 pt. to bushels, &c. Ans. 290 bu. 2 qt. 
8. How many pecks in 42 chaldrons? 

4, Reduce 15 bu. 6 qt. to pints. 

5. How many small measures in 25 bushels ? 

6. At 9 cents a quart, what will a bushel of peaches cost? 


7. How much will a grocer make on 14 bushels of potatoes, 
if he buys them at 75 cents a bushel, and retails them at 12 cents 
a half peck ? Ans. $2.94. 

8. Reduce the following to pints, and add the results: 7 qt.; 
5 bu. 3 pk.; 2 pk. 6 qt. Ans. 426 pt. 

9. Reduce the following to pecks, and add the results: 14 chal. ; 
240 pt.; 19bu.; 186 qt. Ans. 2124 pk. 

10. How many barrels, holding 24 bushels each, will 40 chal- 
drons of coal fill? 

11. Reduce 1879 bu. 3 pk. to quarts. 


TIME. 


267. The natural divisions of time are the year and 
the day. The year is the period in which the Earth 
makes one revolution round the Sun; the day, that in 
which it makes one revolution on its axis. 

The year is divided into twelve calendar months; the 
day, into hours, minutes, and seconds. 


265. What is the standard bushel of the U.S.? How many cubic inches docs it 
eontain? How many wine quarts does a quart of dry measure equal? What is the 
Small Measure ?—266. How is foreign coal imported? How is American coal bought 
and sold ?—267. Name the natural divisions of time. What is the year? What is 
the day? Into what is the year divided? Into what is the day divided ? 


DIVISIONS OF TIME, 155 


TABLE. 
60 seconds (sec.) make 1 minute, . .. min. 
60 minutes, WOM e's 5g jis 
24 hours, DLAs es oy 0s ge SAS 
7 days, 1 week, .... wk. 
365 days or leveay s 
12 calendar months, Tis eNNS 6S 
366 days, 1 leap year. 
100 years, 1 century,. . . cen. 
min. 806, 
h. l= 60 
Be, Les Rares 60 = 3600 
wk, Nhe 24 re 1440 = 86400 
yr. fa ie eelGs © eee C080, y= 604800 
1 = 824 = 3865 = 8760 = 525600 = 315386000 


268. The twelve calendar months, with the number 
of days they contain, are as follows :— 


DAYS. DAYS. 

Ist mo. January (Jan.) 31. | 7th mo. July (July) 31. 
2d mo. February (Feb.) 28. 8th mo. August (Aug.) 381. 
3d mo. March (Mar.) 31. 9th mo. September (Sept.) 30. 
4th mo, April (Apr.) 30. | 10th mo. October (Oct.) 31. 
th mo. May (May) 31. | 11th mo. November (Nov.) 380. 
6th mo. June (June) 380. | 12th mo. December (Dec.) 31. 


269, The days in these months, added together, make 
365 days in the year. But the solar year exceeds this by 
nearly six hours, its exact length being 365 days 5h. 
48min. 49.7sec. To cover this excess, every fourth year 
(except three in four centuries) is made a Leap Year of 
366 days, the additional day being placed at the end of 
February, the shortest month, which then contains 29 
days. Leap Year is also called Bissextile. 

Every year that can be divided by 4 without remainder, as 1868, 1872, 
1876, isa leap year, except the years that are multiples of 100 and are © 


Recite the Table.—268, Name the twelve calendar months in order, with the 
nuraber of days they contain. —269, How many days in these twelve months? What 
is the exact length of the solar year? What provision is made for covering the dif- 
ference between the common and the sular year? What other name is applied to 
Leap Year? What years are leap ycars? 


156 REDUCTION. 


not exactly divisible by 400. The year 1900 wil! not be a leap year, bué 
2000 will be. 

270, In business calculations, 30 days are generally allowed to the 
month. In common language, the term month is often applied to an in- 
terval of 4 weeks. 

The following lines will help the pupil to remember the number of 
days in each calendar month :— é. 

‘‘Thirty days hath September, 
April, June, and November ; 
All the rest have thirty-one, 
Except February alone; 
Which has but four and twenty-four, 
And every leap year one day more.” 


_ 271. The following Table will be found useful :— 


TABLE, 


SHOWING THE NUMBER OF DAYS FROM ANY DAY OF ONE MONTH TO THE SAME DAY 
GF ANY OTHER MONTH WITHIN A YEAR. 


FROM ANY TO THE SAME DAY OF 
DAY OF 


Jan. | Feb. | Mar. | April. May. | June. | July.| Aug. | Sept.| Oct. | Nov. | Dec. 


me 


JANUARY. .|365| 81] 59} 90/120] 151/181 | 212 | 248) 273 | 304 | 334 
Fesruary.| 384/865} 28] 69} 89) 120/150] 181 | 212 | 242 | 273 | 303 
Marcon .../3806;}337/865| 81} 61] 92/122/158 | 184 | 214 | 245 | 275 
APRIL... . .| 275 | 306 | 834/365] 80] 61] 91/122) 153) 183 | 214/244 
May ...../ 245 | 276 | 304/335 |3865| 81] 61] 92)123/153 | 184) 214 
JUNE .....| 214] 245 | 273 | 304) 834/365] 80] 61} 92/122) 153) 188 
JULY .....| 184} 215 | 243 | 274 | 804/835 | 365} 81} 62} 92|128)158 
Aveust...| 153 | 184 | 212 | 243 | 273 | 804} 3384|3865| 31] 61) 92/122 
SEPTEMBER .| 122] 153/181 | 212 | 242 | 273/3038/884/3865| 80} 61] 91 
OctoBer...| 92/123/151)182| 212/248 | 273 | 304| 335] 3865} 81] 61 
NoveMBER.| 61] 92/120|151/181| 212 | 242 | 273 | 304 /|384|365| 30 
DEceMBER.| 81] 62] 90} 121] 151] 182 | 212 | 248 | 274) 304 | 835 | 865 


ExamPLe.—How many days from Nov. 6, 1865, to. the 15th of the 
following April ?—Find November in the vertical column on the left, and 
April over the top. At the intersection of these two lines we find 151, 
which is the number of days from November 6, 1865, to April 6, 1866. 
‘To April 15 will be 9 more days; 151+9 = 160, the number of days 
- required. 

One more day than is given in the above Table must be allowed for 
intervals embracing the end of February falling in a leap year. 


270. In business calculations, how many days are generally allowed to the month? 
To what is the term month often applied in common language ?—271. What doce 
the Table show? Give an example, to illustrate ifs use, 


CIRCULAR MEASURE. 137 


EXAMPLES FOR PRACTIOE. 


1, Reduce 9yr. 3 da. 59 min. to seconds. Ans. 284086740 see. 

2. Reduce 63142980 sec. to years, &e. Ans. 2yr. 19h. 43 min. 

3. How many seconds in a solar year (§ 269)? Ans. 31556929.7 

4, How many leap years from the year 1800 to 1900? 

5. How many days from Apr. 14, 1865, to Dec. 31, 1865? 
(See Table.) To October 9, 1865? To Aug. 29, 1865? 

6. When 41 hours of a day have passed, how many seconds 
remain ? 

7. How much time will a person waste in a year, who wastes 
ten minutes every day ? Ans, 2.da, 12h. 50 min. 

8. If a clock loses 3sec. every hour, how many minutes too 
slow will it be at the end of a week ? Ans. 8 min. 24 sec. 

9. Find the length in days, &c., of the lunar month, which 
-contains 25514438 seconds. Ans. 29 da. 12h. 44 min. 8 see. 

10. If a person’s income is 1c. a minute, what will it amount 
to in the months of June, July, and August ? Ans, $1824.80. 


CIRCULAR MEASURE. 


272. Circular Measure is used in connection with 
angles and parts of circles. 


273, A Circle is a figure bounded by a curve, 
every point of which is equally distant from a point 
within, called the Centre. 
The Circumference of a circle is the curve that A C 
bounds it. A Diameter is a straight line drawn 
through the centre, terminating at both ends in the 
circumference. A Radius (plural radiz) is a straight 
line drawn from the centre to the circumference, D 
and is equal to half the diameter. 
The Figure represents a Circle: ABCD is the circumference; E, the 
sentre; AC, the diameter; EA, EB, EC, are radii. 
An Angle is the difference in direction of two straight lines that meet. 
A Right Angle is an angle made by one straight line meeting another 
in such way as to make the two adjacent angles equal,—that is, so as to 
meline no more to one side than the other. In the above Figure, BEA 
and BEC are right angles. 


B 


272. With what is Circular Measure used in connection ?—273, What is a Cirele: 
What is the Circumference of a Circle? Whatisa Diameter? Whatis a Radius} 
What isan Angle? What is a Right Angle? 


. Pa 4 a 4 Sa « . - * 


158 REDUCTION, 


974, Every circle may be divided into 360 equal parts, 
called Degrees. The actual length of the degree will of 
course depend on the size of the circle. A degree is 
divided into 60 equal parts, called Minutes ; and a minute 
into 60 equal parts, called Seconds. 


TABLE. 

60 seconds (") make 1 minute, . . .’ 
60 minutes, Lidegrees sis 8 33- set. 
30 degrees, LeRLOT ates as Meese 
12 signs, LCI 1% deere oO! 
1 60’ 

8. ees 60 -= 3600 

C. De S02 e800 3 S000 
fee ol ae OU Ss U0 ee DUOe 


275, The Sign is used only in Astronomy.—1 minute of the circum. 
ference of the earth constitutes a geographical or nautical mile, which, as 
we have seen, is about 134, statute miles. 


-~EXAMPLES FOR PRAOTIOCOE. 


1. How many seconds in } of a circle? Ans. 324000’. 
2. Reduce 40° 41’ 42” to seconds. Ans. 146502”. 
3. Reduce 251989” to degrees, &c. Ans. 69° 59’ 49”. 
4. How many minutes in two signs? 

5. How many geographical miles in 5° of latitude? 


: 276, PAPER. 


24 sheets make 1 quire. 


20 quires, 1 ream. 
2 reams, 1 bundle, 
5 bundles, 1 bale. 
quire, sheets. 
ream. = 24 
bundle. Lp se Oo =e et 480 
bale. LAasess de2ani as Gedy eae OGD 
Lee Oe lO -oee ce 200F ee 4806 


274. Into what may every circle be divided? How isa degree divided? How 
is a minute divided? Recite the Table of Circular Measure.—275. In what alone is 
the Sign used? What does 1 minute of the circumference of the earth constitute f 
—276. Recite the Table used in connection with paper, 


REDUCTION OF DENOMINATE FRACTIONS, 159 


277, COLLECTIONS OF UNITS. 
12 units make 1 dozen, doz. 


12 dozen, 1 gross. 
12 gross, 1 great gross. 
20 units, 1 score. 
doz. units. 
gross, 1 = 12 
great gross, 1 = 12 = 144. 


EXAMPLES FOR PRAOTICE. 


1. How many sheets in 10 bundles of paper? 

2. If paper is $6 a ream, what does it cost a quire? 

3. A bookseller bought 10 reams of paper, at $24 a ream; he 
retailed it at 1 cent a sheet. What was his gain? Ans. $23. 

4. How many reams of paper will be needed for 1000 books, 
if each book requires a dozen sheets? Ans. 25 reams, 

5. Ifa score of boys have each 5 boxes of pens, containing a 
gross apiece, how many pens have they in all? 

6. A tailor uses 13 dozen buttons out of a great gross; how 
many buttons has he left ? | a 

7. If a stationer manufactures 48 dozen copy-books a day, 
excluding Sundays, how many great gross will he make in fifty- 
two weeks? Ans. 104 great gross. 


Reduction of Denominate Fractions, 
Common and Decimal. 


278. A Common Fraction or Decimal is called De- 
nominate when it is used in connection with a denomina- 
tion; as, £4, .25 oz, 

279. Denominate Fractions, whether common or deci- 
mal, are reduced, like integers, to lower denominations 
by multiplication, to higher denominations by division. 


277. Recite the Table relating to collections of nnits.—278, When is a common 
fraction or decimal called @enominate?—279. How are denominate fractions reduced 
to lower denominations? To highcr denominations ? 


160 REDUCTION OF 


280. Casz L—Zo reduce one denominate fraction to 
another of a lower denomination. 
Exampie.—Reduce 74, gall. to the fraction of a gill. 


This is a case of Reduction De- 1 A 9 A 2 
seending. Multiply the given fraction -__ yxy ~ x © x 2 = 
by 4 (since 4 qt. = lgal.); by 2 (2pt. 42 1 1 1 7 
=1qt.); by4(4gi. = 1pt.). Cancel $8 7 
such factors as are common, and multi- 

ply together those that are left. 


Rutz.— Multiply the given fraction by the number or 
numbers that connect tts denomination with that of the 
required fraction. 


Ans. 2 gill, 


EXAMPLES FOR PRAOTIOER. 


1. Reduce 545, ton to the fraction of an oz. Ans. $02. 
2. Reduce £3,;5 to the fraction of a penny. Ans, 7d. 
3. What fraction of a pint is 7,4 of a bushel ? Ans. $4 pt. 
4, What part of a sq. foot is s¢25n5 acre ? Ans. sol5 Sq. ft. 
5. What part of an inch is sp@soo Of a mile? Ans, $34 in. 
6. What part of a second is gr@pay Of a Week? = Ans. $$ ec. 
7. What part of a quire is 7%, of a bundle of paper ¢ 

8. Reduce +4, of a pound to the fraction of a scruple. 


281. Case IL—TZo reduce a denominate fraction to 
whole numbers of lower denominations. 


ExaMPLe.—Reduce 2 of a bushel 2 
to pecks, &c. = 


To reduce bushels to pecks, multiply by 4. ke) vba 
Multiplying the numerator of the fraction by 2 pk. |2 rem. 
4 and dividing the product by its denominator, 8 
we get 23 pk. Reduce the fraction, 2 pk., to 3) 16— 
quarts. Multiplying its numerator by 8 and ae 
dividing by its denominator, we get 54 qt. 5 qt. | 1 rem. 
Reduce the fraction, } qt., to pints. Multi- 2 
plying its numerator by 2 and dividing by its 8)2 
denominator, we get pt. Collect the integers O32 vt 
in the several quotients, and the last fraction, apt. 
for the answer. Ans. 2 pk. 5 qt. 3 pt. 


280. What is the first Case of the reduction of denominate fractions? Solve the 
given example. Recite the rule.—281. What is Case II.? Go through the giver 
example. 


DENOMINATE FRACTIONS. 161 ~- 


Ruitzt.—Multiply the numerator of the given fraction 
by the number that will reduce tt to the next lower denomi- 
nation, and divide by its denominator. If there ts a re- 
mainder, multiply and divide tt in the same way ; and 
proceed thus to the lowest denomination. Collect the 
integers and the last fraction, tf any, for the answer, 


EXAMPLES FOR PRACTICE. 


Reduce the following to integers of lower denominations :— 


1, 8 of a pound Troy. Ans. 7 oz. 4pwt. 
2. $ of a sign. ; Ans. 22° 30’. 
3. 44 of a cubic yard. Ans. 19 cu. ft. 138822 cu. in. 
4, =, of a bar. (beer measure). Ans. 32 gal. 1 qt. 14 pt. 
5. + mile (surveyors’ measure). Ans. 45 ch. 71 li, 3.393 in, 
6. 3 of a great gross. Ans. 7 gross 6 dozen. 
7. zh,y of a hundred-weight. Ans. 12 oz. 124 dr. 
8. 3%; of a long ton. Ans, 268 Ib. 12 oz, 124 dr. 
9. 2 of a furlong. 


10. 2 of a shilling. 

11. How many acres, &c., in a piece of land 4 mile long and 4 
of a mile wide? Ans. 142 A. 853 sq. rd. 

Area'= 2 x 4 = 2:sq. mi. Reduce 2 sq. mi, to acres, &c. 

12. Required the solid contents of a block of stone, 24 yd. long, 
14 yd. wide, # yd. thick. Ans. 1cu, yd. 21 cu. ft. 10364 cu. in. 


282. Casz III.—Zo reduce one denominate fraction te 
another of a higher denomination. 


ExampLe.—Reduce 2 of a gill to the fraction of a 
gallon. 


This is a case of Reduction Ascend- 2 1 
ing. Divide the fraction: that is, mul = See Sane 
tiply its ghee by 4 (since4gi, 7 x 4x 2X 4 112 
= 1pt.); by 2 (2pt. = 1qt.); by 4 
(4 qt. a) 1 gall.). Cancel 2; ye i Aas th gall 
the remaining factors. 

Under Case I. we reduced ;1+5 gall. to ? gill. Here we have reduced 


Recite the rule for reducing a denominate fraction to whole numbers of lower 
denominations.—232. What is Case IT].? Solve the given example. Mow may it be 
proved? 


162 REDUCTION. 


% gill to ;4y gall. Hence the operations in Case I. and Case III. prove 
cach other. 


Ruitz.— Divide the given fraction by the number or 
numbers that connect tts denomination with that of the 
required fraction, 


EXAMPLES FOR PRACTICE, 


1. Reduce $ of a rod to the fraction of aleague. Ans. 33,¢ lea. 

2. Reduce ;'; pt. to the fraction of apuncheon. Ans. 55 pun. 

3. Reduce 4 fathom to the fraction ofa mile. Ans. »¢p5 mi. 
4 fathom = 2feet %x4tx7xaxt=rdns 

4, What part of a guinea is 4 of a crown? Ans. 34 guin. 

5. What part of an eagle is 4 of a dime? Ans, xi, E. 

6. What part of a long ton is 2 of a pound? 

7. What part of a pound is 34 of a scruple? 

8. What part of a circle is 2 of a second? 

9. What part of a piece of 40 yards is a nail of cloth? 


1 nail = Te yd. as Mp SS eit Ans, 
10. What part of 20 gallons is 19 of a pint? Ans. +> 
11. What part of a five-acre lot is $ of a perch ? Ans. yh>- 


12. What part of the month of Aug. is 7 min.? Ans. syq4535- 


283. Case IV.—7o reduce one denominate number to 
the fraction of another. 


Examrpie I.—Reduce 16s. 6d. 2 far. to the fraction of 


a pound, 
Reduce 16s. 6d. 2 far. to farthings, the 

lowest denomination mentioned : 16s. 6d. 2 far. = 794 far. 
Reduce £1 to the same denomination : £1 = 960 far. 
794 far. = ee of 960 far. 3 LT24 = £397 Ans. 


Reduce this fraction to its lowest terms. 
Exampte II.—Reduce 20 rods 24 yards to the fraction 
of a mile. 


If the lowest denomination given contains 4, we must reduce both 
numbers to halves of that denomination ; if it contains thirds, to thirds, 


Give the rule for reducing a denominate fraction to a higher denomination.— 
283. What is Case 1V.? Solve Example I. If the lowest denomination given con- 
tains }, what must we do? If it contains thirds, wha. must we do? Illustrate this 
with Example IL, 


REDUCTION OF DENOMINATE DECIMALS. 163 


&c. In this example, for instance, we must reduce both numbers to half- 
yards, 
20rd. 24 yd. = 225 half-yards, 
1 mile = 3520 half-yards. 
#25 = ziic mile Ans. 

Rutr.—Reduce the given numbers to the lowest de- 
nomination in either. Of the numbers thus reduced, take 
the one of which the fraction is required for the denomi- 
nator, and the other for the numerator. 


EXAMPLES FOR PRAOTIOE. 


Reduce the following; give the fraction inits lowest terms :— 


1. 8 bu. 1 pk. to the fraction of a chaldron. Ans. 44 chal. 
2. 1 0z. 1 pwt. 1 gr. to the fraction of a lb. Ans. 31°!5 Ib. 
3. 53 0z. to the fraction of a stone. Ans. giz stone. 
4, 34cu. ft. to the fraction of a cord. Ans. 2; cord. 
5. + inch to the fraction of a hand. Ans. 3; hand. 
6. 29 gal. 1 pt. to the fraction of a barrel. Ans. $33 bar. / 
7. 1 English ell to the fraction of 1 French ell. Ans. 3 ell Fr. 
Reduce both to the common denomination, quarters. 
8. What part of 1 ch. 501. is 44 inches? Ans. zits. 
9. What part of 6s. 83d. is 3s. 5d. ? Ans. 4$4. 


10. Reduce 54 hours to the fraction of a leap year. 


284, Cast V.—TZo reduce a denominate decimal to 
whole numbers of lower denominations. 


ExampLtr.—Reduce .471875 Ib., 471875 Ib. 
apothecaries’ weight, to ounces, &c. 12 

This is a case of Reduction Descending. oz. 5 | 662500 
Multiply by 12, to reduce to ounces, pointing wot 3 
off the product as in multiplication of deci- dr. 5 | .800000 


mals. Reserve the integer, and reduce the 
decimal to drams by multiplying by 8. Again 


reserve the integer, and reduce the decimal to se. .900000 
scruples by multiplying by 3. There being no 20 
integer, multiply this product by 20 to reduce gr. 18.000000 


it to grains. Finally, collect the integers in 
the several products for the answer. Ans. 5 oz, 5 dr. 18 gr. 

Recite the rule for reducing one denominate number to the fraction of another.— 
284. What is Case V.? Go through the given example, explaining the steps, 


164 REDUCTION. 


RvuiLe.— Multiply the given decimal by the number that 
will reduce it to the next lower denomination. Treat the 
decimal part of the product in the same way, and pro- 
ceed thus to the lowest denomination. Collect the integers 
in the several products, with the last decimal, if there is 
one, for the answer. 


EXAMPLES FOR PRAOTIOE. 


- Reduce .725 Ib. Troy to ounces, &c. Ans. 80z, 14 pwt 

. Reduce .4156 cwt. to qr., &c. Ans. 1 qr. 161b. 8 oz. 15.36 dr. 

. Reduce .75 bale of paper. Ans. 3 bundles 1 ream 10 qui. 

. Reduce .9 of a great gross to gross, &c. 

. Reduce .002 bar. of beer to gallons, &c. Ans. .576 pt. 
6. A lot is 50.3 rd. long, 29.25rd. wide. What is its area in 

acres, &c. ? Ans. 9 A. 81 sq.rd. 8 sq. yd. 2 sq. ft. 125.1 sq. in. 


Area = 50.8 x 29.25 = 1471.275 sq. rd. Reduce 1471 sq. rd. to roods and acres. 
Reduce .275 sq. rd. to square yards, &c. Combine the results. 


7. A cistern is 3.25 ft. long and wide, and 10 ft. deep. What 
is its capacity ? Ans. 3 cu. yd. 24 cu. ft. 1080 cu. in. 
8. A piece of land measures 32.72 ch. by 41.36ch. Required its 
area in acres, roods, and perches. Ans..135 A. 1R. 12 perches+. 


Area = 82.72 x 41.36 = 1353.2992 sq. ch. Dividing 1353.2992 sq. ch. by 10 
(since 10 sq. ch. = 1 acre), we get 135.82992 acres. Reduce .32992 A. to roods and 
perches. 


9. What is the area of an oblong field, 8.5 chains inlength and 
5.5 chains in width? Ans, 4A. 2R. 28 sq. rd. 

10. How many degrees, &c., in .01 of a circle? 

11. How many days, &c., in .12 of a year? 

12. How many roods, &c., in .575 of an acre? 

13. How many shillings, &o., in .49 of a pound sterling ? 


Cr HH 9 DO BR 


285, Case VI.—Zo reduce a compound number to the 
decimal of a higher denomination. 

ExamMpLe.—Reduce 5 oz. 5 dr. 18 gr. to the decimal of 
a pound, 


Recite the rule for reducing a denominate decimal to whole numbers of lower 
{enominations.—285, What is Case VI.? Solve the given example. 


REDUCTION. 165 


Begin with the lowest denomination. Reduce 


18 gr. to the decimal of a dram, which is the next 60) 18.0 gr. 
higher denomination given, by dividing by 60 .odr. 
(since 60 gr. = 1 dr.), annexing as many decimal 8)5.3d 
naughts as may be necessary. Annex the result, ) 5.3 dr. 
.3dr., to the drams in the given number, and -6625 oz. 
divide by 8, to reduce to the decimal of an ounce. 12) 5.6625 oz. 


Annex the result, .6625 oz., to the ounces in the —____.— 
given number, and divide by 12, to reduce to the Ans. 471875 Ib. 
decimal of a pound. 

The processes in Case V. and Case VI. prove each other :— 

By Case V.. .471875 lb. = 502, 5dr. 18 gr. 
By Case VI. 50z 5dr. 18gr. = .4718765 lb. 

Rurz.— Divide the lowest denomination by the num- 
ber that will reduce it to the next higher denomination in 
the given number, and annex the decimal quotient to that 
next higher. Treat this result in the same way, and pro- 
ceed thus till the required denomination is reached. 


EXAMPLES FOR PRAOTIOCOE. 


Reduce the following; prove the answers :— 


. 2cwt. 3 1b. to the decimal of a ton. 

. 1pk. 7 qt. 1 pt. to the decimal of a bushel. 

10. 10 25 to the decimal of a pound. 

11. 7 dr. 18 gr. to the decimal of an ounce. 

12. 8 pwt. 3 gr. to the decimal of an ounce. 

13. 16 rd. to the decimal of a mile. Ans. .05 mi. 
14. 8 in. to the decimal of a fathom. Ans. .i fathom. 
15. 1 Ib. 12 oz. to the decimal of a stone. Ans. .125 stone. 
16. 24 1b. to the decimal of along ton. Ans. .010714 long T.+. 


1. 2gal. 2 qt. 1 pt. to the decimal ofa hhd. Ams. .0416 hhd. 
2. 3s. 44d. to the decimal of a pound. Ans. £.16875. 
3. $5.10 to the decimal ot a double eagle. Ans. .255- 
4. 2da. 3h. 4min. 6sec. to the decimal of a week. 

5. Ted. ft. 7 cu. ft. to the decimal of a Cd. Ans. .9296875 Cd. 
6. 4 yd. 9in. to the decimal of a rod. Ans. .772 vd. 
7. 9d. 2far. to the decimal of a crown. Ans. .1583 crown. 
8 

9 


Recite the rule for reducing a compound number to the decimal of a higher do- 
nomination, 


166 REDUCTION. 


MISCELLANEOUS QuEstTions.—In what denominations do American mer- 
chants keep their accounts? British merchants? What American coin 
is nearest in value to the British shilling? To the British sovereign? 
Why is Federal Money so called? Sterling Money? 

Recite the three Tables used in connection with weight. For what is 
Avoirdupois Weight used? Apothecaries’? Troy? In which of these 
is the pound the greatest? In which is the ounce the greatest? Is the 
avoirdupois dram greater or less than the dram of apothecaries’ weight? 
Is the grain of apothecaries’ weight greater or less than the Troy grain? 
How many pennyweights is the dram of apothecaries’ weight equal to ? 

What measure is used in reckoning distances? In surveying land? 
In expressing superficial contents? In expressing solid contents? In 
estimating the amount of work in solid masonry? In estimating surfaces 
to be plastered or paved? In measuring drygoods? What are the di- 
mensions of a cord of wood? 

What measure is now generally used for liquids? How are the con- 
tents of casks ascertained ? How many cubic inches in the wine gallon ? 
In the beer gallon? Which is greater, the beer or the wine quart? What 
is used in measuring grain and fruit? Which is greater, the quart of 
dry or that of liquid measure? In what two Tables do the second and 
minute occur? How do thesecond and minute of Circular Measure differ 
from those of Time Measure ? 


986. MIscELLANEOUS EXAMPLES. 


1. How many ducats, worth 9s. 3d. apiece, are equal in value 
to £74? Ans, 160 ducats. 

2. If a cannon-ball could move with uniform velocity 1000 
feet a second, how many miles, &c., would it go in q quarter of a 
minute ? a‘ 

3. How long would this ball be in reaching the sun, which is 
95000000 miles from the earth ? Ans. 5805 da. 13 h. 20 min. 

4, A cubic foot of water weighs 1000 oz. What weight of 
' water will a cistern 3 ft. by 4 ft. across, and 10 ft. deep, con- 
tain ? Ans. 75 cwt. 

5. Required the area in acres, &c., of an oblong piece cf iand, 
.5 miles long and .3 miles broad. 

6. If three presses, each capable of striking off 1806 coins an 
hour, work, tho first at querter-dollars, the seccnd at half-eagles, 


MISCELLANEOUS EXAMPLES. 167 


and the third at dimes, what will be the whole amount coined in 
eight hours ? Ans. $77040. 
7. A silversmith, having on hand 20 Ib. of silver, uses 4 oz. 
18 er. of it. What decimal is this of the amount he originally 
had ? | Ans. .0168229-+. 
Find what decimal it is of 1 Ib., § 2853; it will be 3, as much of 20 Ib. 
8. What were the solid contents of the Ark, which was 300 
cubits in length, 50 in breadth, and 30 in height—the sacred cubit 


being 22 inches ? Ans. 102700 cu. yd. 16 cu. ft. 1152 cu. in. 
9. In two dozen bottles, each holding 1.1 qt., how many gal- 
lons, &c. ? Ans. 6 gal. 2 qt. 3.2 gi. 


11x24 = 26 .4qt. Reduce 26 qt. to gallons, and .4 qt, to lower denominations 
according to § 2S4. 


10. An oblong piece of land measures 14 ch. 5]. in width, and 
86 ch. 241. in length. How many acres, roods, and perches, does 
it contain ? Ans. 50 A. 8R. 26.752 P. 

11. What part of an acre is an oblong lot 75 feet wide and 150 
feet in length ? Ans. 125 A, 

12. What are the solid contents of a block of wood, # ya] toh 
yd. wide, 2 yd. thick? Ans. 4 cu. ft. 1086.8 cu. in. 

13. How many acres, &c., are there in an oblong TAT rp 4 mi 
long, # mi. wide? 

14. If 1 of a chaldron of coal is consumed daily, how many 
bushels will be used in a week ? 

15. If a thread 18 rods long can be spun from an ounce of 
silk, how many pounds of silk will be required for a thread 90 
miles long ? | Ans. 100 Ib. 


16. Reduce 3 qt. to the decimal of a bushel. 
=.15 qt. .75+8=.09875 pk. .09875 + 4 = .0284875 bu. 
17. Reduce 4 qt. to the fraction of a hhd. Ans. x45 hhd. 


To the fraction of a pint. Ans. 1 pt. 
To lower denominations. Ans. 1 gi. 
To the decimal of a gallon. Ans. .03125 gal. 
18. Reduce 13 145 to the fraction of a Ib. Ans. +; Ib. 
To the decimal of an ounce. Ans. .2 0Z. 


19. Reduce 5 sq. rd. to lower denominations. 


168 COMPOUND ADDITION. 


CHAPTER XIV. 
COMPOUND ADDITION. 


287. Compound Addition is the process of uniting two 
or more compound numbers in one, called their Sum. It 
combines addition and reduction ascending. 


ExampLE.—aAdd 1 lb. 3 02. 19 pwt. 23 gr.3; 2 oz. 15 gr.; 
3 lb. 17 pwt.; and 2 lb. 1 oz. 8 pwt. 10 gr. 


That we may unite things of the same kind, we write pounds under 
pounds, ounces under ounces, &c., marking the denominations above. 
Beginning to add at the right, we find the sum 


of the grains to be 48. But 48¢r.— 2 pwt. Hence Tec Qt Dib BM 


we write 0 under the grains, and add the 2 pwt. in 1 3 19 23 
with the other pennyweights. 2 A) 15 

The sum of the pennyweights is 46. But 46 3 0 17 O 
pwt. = 2 oz. 6 pwt. Write 6 under the penny- 2 1h 8518 


weights, and add 2 to the column of ounces. CPR EY 
The sum of the ounces is 8, which, not being AM PES 0 
reducible to pounds, we write under the ounces. The sum of the pounds 
is 6, which, not being reducible to any higher denomination, we write 
under the pounds added. Ans. 6 Ib. 8 oz. 6 pwt. 


288. Observe that in Simple Addition there is a similar reduction, 
when the sum of a column exceeds 9. As the orders increase in value 
tenfold as we go to the left, to reduce to a higher order, we divide the 
sum of each column by 10. That is, we cut off the right-hand figure, 
and place it as a remainder under the column added; while the left- 
hand figure or figures, being the quotient, we add to the next column. 


289.—RvuLp.—1. Write numbers of the same denomi- 

nation in the same column. 

2. Beginning at the right, add as in simple numbers. 
Write each sum under the numbers added, unless it can 
be reduced to a higher denomination ; in which case, 
divide by the number that tt takes to make one of that 
denomination. Write the remainder under the numbers 
added, and carry the quotient. 

3. Prove by adding in the opposite direction. 


287. What is Compound Addition? What processes does it combine? Go 
through the given example, explaining the steps.—288. Show how in Simple Addi- 
tion there is a sinailar reduction.—289, Recite the rule for Compound Addition. 


EXAMPLES FOR PRACTICE, 


990.—If a fraction occurs in the an- 


169 


swer, it must be reduced to lower denomi- pra a ft. 
nations, if there are any, and the result yt 
added to the previous sum with the frac- Add, 5 <2 1 
tion omitted. Thus, in dividing yards by 6 3 2 
54, to reduce them to rods, a remainder ap ltck 
containing 4 yd. may occur, as in Example 19 13/1 
2. But byd = 1ft. 6 in, We therefore jyq, — 4 Gin 
add 1 ft. 6 in. to the integers of the answer —— 
first obtained. Ans. 19. 1 2 6 
291. ExampLtes For PRraoTiokr. 
Add the following compound numbers :— 
(1) (2) (8) 
x d. far. Dar 2) 25: ). ® . or. rd... xd, ft. in. 
S976, ra TM? 37 2 418 19, 5. “324 
8 Oe Oconee bee ree 2) oe Ulises 
t eae! eeeys 6 Nees 3 4 OW 1G Bip Qn y 2 
(eye fy peas GaSe Be ee Ba Ag! te ity, 
2 eGo 412 Bee Qe ee LiPa Se srk 
Sak 2 0 OS 1 ee Pere 19 39 2733 Oe 89 
(4) (5) (6) 
ch. ik in. sq.rd. sq. yd. sq.ft. sq. in. Cd. ed. ft. 
9 41 675 Ge Boren a 93 4 2 
14 ti Seo oUt 56 Seah 
8 57 38 Be TPS Ot OT 32. 8 
O21 G16. ;i) 1 385 82006 79 15 7 
35 82 4,3, 144 14.8 128 29 8 
90 (EP RS 39 3800) oI 91 P Oaaids 
@) (8) (9) 
gal. gt. pt. gi. TT. -ewt.. qr! Ib oz stone Ib. 0% 
Inge orbsiat3 203 toPPoon1 & 45 9 HOMITR: 4 
2. hen 41 16 0 4 5 Tt sdLyicok 
pli gkkeemee 1M a Ds Sa bs aa A. lay 52 
9243." SO ee O et De Pe be deve. SUL e 


10. A jeweller buys the following quantities of silver: 8 Ib. 6 
pwt.; 100z. 4pwt. 21 gr.; 80z. 20gr.; 3]b, 6oz. S8pwt. 7 gr. 


How much does he buy in all? 


290, If a fraction occurs in the answer, what must be done ? 


8 


170 COMPOUND ADDITION. 


(11) (12) 

euyd. cu.ft  cuin, mf tute Pe Ve oo ft in, 

23 19 16984 AT lpr 29a 45 0 6.6 

48 22 8425 Q6Hr ON ABs Wd D 3.1 

"9 8 19574 BOA Beg. eiBeeT edd 

52. eas 2083 SHUT CB6t. TIO EG 

87 14 12654 34 6 33 4. | ed 

65 16 108428 ae ee Se ee CO 

85%. 19 -431173 DAT 30) too Oe 4.10 

(18) (14) (15) 
bu pk. qt: Sept. Wk. da. 4h. “min? “sée: e “ & 
LAS te OL 2 2 5 PAR 286 8 36 24 
Dono aso we Wyk iiss a) pues abate No! 4 8 14 
EAN CES Sas Ia) 3 Le ES 8 2929 6 9 36 
6. 42 4 G2 § Ss 2 FOR 59 Dy Die 
gees ae i 2 Be) 8 4 Wt @bTt BT 2° 5f 42 


16. What are the contents of four hogsheads, the first of which 
contains 63 gal. 2 qt. 14 pt.; the second, 60 gal. 3 qt. 1.75 pt.; the 
third, 62 gal. 1 pt. 3 gi.; the fourth, 61 gal. 2 qt. 2 gi.? 

Ans. 248 gal. 1 qt. 1 pt. 2 gi.. 

17. Ilow much wood in three piles, the first of which contains 
10 Cd. 6 ed. ft. 4 cu. ft.; the second, 12 Cd. 12 cu. ft.; the third, 
17 Cd. 1 ed. ft. ? 

18. A surveyor measures four distances; the first he finds to 
be 40 ch. 591. 3in., the second 28 ch. 481. 5 in., the third 16.27 
ch., the fourth 12 ch. Tin. What is the whole distance measured, 
expressed first in chains, &c., then in the denominations of linear 
measure ? 

Ans. 97 ch. 801. 7.08in.: 1 mi. 1 fur. 29rd. 1 yd. 10.68 in. 

19. How many yards in 8 pieces of cloth, containing respec- 
tively 24 Ells French 3 qr. 1 in., 28 Ells English 8 qr. 2 nails, 40 
Ells Flemish 1 qr. 1 nail 14 in. ? 


2} inches make 1 nail; 4 nails, 1 qr. of a yd.; 8 qr.,1 Ell Flemish; 5 qr.,1 Ell 
English; 6 qr, 1 Ell French. Reduce the ells to the common denomination, quar- 
ters; add the whole, and reduce the quarters to yards. Ans. 103 yd. 


20. Find the sum total in pounds, &c., of the following 
items: £20 10s., £1 63. 8d., 5 guineas 10s. 6d., 15 guineas, and 
£1 15s, 33d. Ans. £45 2s, 54d, 


COMPOUND SUBTRACTION. 171 


21. A person owning a section of land ($251) buys three addi- 
tional tracts, containing 347 A. 2R. 27 sq. rd., 201 A. 19 sq. rd., 
and 417 A. 3R. 14sq.rd. How much does he then own m all? 

Ans, 2sq. mi. 826 A. 2 R. 20 sq. rd. 

22. How much coke in three carts, the first of which contains 
1 chal. 5 bu. 2 pk., the second 1 chal. 54 bu., and the third 35 bu. 
8 pk. ? 

23. How much beer in four hogsheads, containing respec- 
tively 538 gal. 2 qt., 54 gal. 1 qt. 1pt., 52 gal. 3 qt. 1 pt., and 51 gal. 
3 qt. 1 pt.? 

24. Add together 6 da. 37 min., 48 da. 5h. 29sec., 94 da. 19h. 
18 sec., 126 da. 7h. 9 min. 8 sec., and 94da. 16h. 13 min. 5 sec. 
How many years in the sum? 

25. How many yards in four pieces of cloth, containing re- 
spectively 30 yd. 1 qr., 20 Ells Fr. 1na., 24 Ells En. 14in., 32 Ells 
Fl. 2 qr. 2na. $in. Ans. 115 yd. 


CH Atel iy he cal 
COMPOUND SUBTRACTION. 


292. Compound Subtraction is the process of finding 
the difference between two numbers, when one or both 
are compound. 


EXaMPLe 1.—From 20 Ib. 5 oz. 3 dr. take 18 lb. 7 0z. 1 dr. Write the 
subtrahend under the minuend, pounds under pounds, &c., marking the 
denominations above. Begin to subtract at the 
right. 1dr. from 8 dr. leaves 2 dr., which we Ib. oz. dr. 
write in the column of drams. 20 ra 

7 oz. can not be taken from 50z. We there- 18 (eed 
fore take one of the next higher denomination yy, 1 14 2 
(11b.), reduce it to ounces, and add it to the 5 0z.; ; 
16 +5 = 21. Then subtracting 7 from 21, we get 14, which we write 
under the ounces.—To balance the 16 oz. added to the minuend, we now 


292. What is Compound Subtraction? Go through the given example explain- 
fine the steps, 


172 COMPOUND SUBTRACTION. 


add 1 Ib. to the subtrahend; 19 lb. from 20 1b., 1 Ib. Answer, 1 Ib. 14 
oz. 2 dr. 


This process involves the same principle as carrying in Simple Sub- 
traction. In the latter, as the orders uniformly increase in value tenfold, 
we add 10 to the figure of the minuend when it is necessary, and to bal- 
ance it add 1 to the figure of the next higher order in the subtrahend. 

293, Rutu.—l. Write the subirahend under the min- 
uend, placing numbers of the same denomination in the 
samecolumn. Beginning at the right, subtract as in 
simple numbers. 

2. If, in any denomination, the subtrahend exceeds 
the minuend, add to the latter as many as make one of 
the next higher denomination. Subtract, and add 1 to 
the subtrahend in the next higher denomination. 

3. Prove by adding remainder and subtrahend. 

We may have to add 1 to the next higher denomination of the sub- 
trahend several times in succession. Thus, 
in Example 2, 53 yd. can not be taken from EXAMPLE 2. 
4yd, Add, therefore, to the minuend 5} yd., mi, fur, rd. yd, 
which equal lrd. 4 + 54 = 9$. Subtracting Brom 1 et ae 
53 from the sum, we get 33 yd., and adding Tak 7” 39 52 
1 successively to the columns of rods, fur- BRO a eee Fae 
longs, and miles, we find the remainder tobe Rem. 0 O O 38 
0 in each case. Ans. 33 yd. , 

If a fraction occurs in any denomination of the remainder, except the 
lowest, it should be reduced and added, as in Addition, § 290. 

294, To find the interval between different dates since 
the Christian era, Write the earlier date under the later, 
representing the month in each by its number (January, 
1; February, 2, &c.). Subtract, allowing 30 days to the 
month and 12:months to the year. 

Examete 8.—Washington was born Feb, 22, 1732. 


How old was he July 4, 1776? 


Represent July, the seventh month, by 7— ret ee 
and February, the second month, by 2. Thirty Nixie Tons 7 a: 
days being allowed to the month, we subtract 1782 2 22 
22 from 80 + 4, and carry 1. Ansan tate els 


Show how tke same principle is involved in carrying in Simple Snbtraction.—~ 
293. Recite the rule. If fractions occur, how are we to proceed? Illustrate this 
with the given example.—294, Give the rule for finding the interval between differ- 
ent dates since the Christian era. Apply this rule to Example 3, 


COMPOUND SUBTRACTION. 173 


EXAMPLES FOR PRAOTIOE. 


(1) (2) (5) 
gal. qt Die ei : f us bit.,.pk. 4 qtr to ptt 
Fropigedge les, “len 3 8. in OS oe LE To 8 te) Lee 
Take 0 mga igri degaleay xy 0. oF g Salutes PAN Fae | 
Ans gso le Ot Oe aon Zon ot. See 
(4) (5) 
mi, piur: 2rd. yds. ft.fin: A. RR. P. sq. yd. sq. ft. sq. in. 
BiB er Oe 148 9.20 80.1: eh 26 
Dorie o0nn be 0 Q 3 ee [goes eosin 
oh She 41 7 8 ON PEE eee Bee 
1 6=tya 3 2 36 =18q. yd 
2 odo te O ras OLA AS ae One) baer od cl eer G TUN nies 
(6) (1) (8) 
ch. Teer 307: x 6. d. nhhd. “bars gall qt... pt. 
20 Sts 6 5 104 4 1 0% <b 20 
16.; 17+ 44 537 12 S$ Qaershd -Bl*% +135 el 
3 00 682 13.14 106 Hla he8 ea 


Find the value of the following. Prove each example. 

9. 25° 15’ 81” — 18° 52’ 49”. 

10. 20 guineas — 19s. 11d. 2 far. Ans. 19 guin. 1s. 2 far. 
11. 1 mi. — 47 ch. 941. 64 in. Ans. 82 ch. 51. 1.42 in, 
12. 15 Cd. 4cd. ft. — 10 Cd. 132 cu. ft. 

13. 30 gal. 3 qt. — 24 gal. 1 pt. 2 gi. 

14. 200 da. 13h. 15 sec. — 195 da. 21h. 49 min. 

15. 9lb. 30z. 1sc. 19 gr. — 8lb. 2dr. 2sc. 6 gr. 

16. 2 T. 9ewt. 8lb. 4dr. — 3 qr. 24]b. 15 oz. 13.dr. 

17. 8wk. 10h. 11 min. — 1 wk. 6 da. 49 min. 57 sec. 

18. 6b. 30z. 15 pwt. 15 gr. — 4]b. 10 oz. 18 pwt. 22 gr. 

19. 9sq. mi. 8sq.rd. 8 sq. ft. —1R. 29 sq. yd. 100 sq. in. 

20. 11 cu. yd. 111 cu. in. — 8 en. yd. 20 cu. ft. 10004 cu. in. 


* Wine Measure. Carrying 1, we get 32 gal., which can not be taken from one 
barrel reduced to gallons (314 gal.). ILence we take two barrels, reduce to gallons, 
subtract, and earry feo. 314 x 2 = 63. 63-382 = 81, Remember, in such a ease, 
to carry 2, 

D 


174 COMPOUND SUBTRACTION, 


91. Henry Clay died June 29, 1852, aged 75 yr. 2 mo. 17 days. 
When was he born? Ans. April 12, 1777. 

22. Andrew Jackson was born March 15, 1767; he became 
president March 4, 1829. What was his age at that time? 

23. Shakespeare was born April 23, 1564. How long from 
that time to the first day of the present year ? 

24. How old was Shakespeare at the time of Milton’s birth, 


December 9, 1608? Ans. 44 yr. 7mo. 16 da. 
25. A note dated Dec. 30, 1862, was paid Nov. 3, 1865. How 
long had it run? Ans. 2yr. 10 mo. 3 da. 


26. San Francisco is in 122° 23’ west longitude, Baltimore in 
76° 37’ west; what is their difference of longitude? Ans. 45° 46’. 
27. Longitude of Boston, 71° 3’ 58” W.; of Rome, 12° 28’ 40” 
E. What is their difference of longitude ? Ans. 83° 82’ 88”. 

The one being in west long., the other in east, to find the diff. of long., add. 

28. New York is in 40° 42’ 43” north latitude; New Orleans, 
_ in 29° 58’ N.; Charleston, in 32° 46’ 33” N. What is the differ- 
ence of latitude between New York and New Orleans? Between 
New York and Charleston? Between Charleston and N. O.? 

29. Latitude of St. Louis 88° 27’ 28” N.; of Cape Horn, 55° 
58’ 40’ S. What is their difference of latitude? Ans. 94° 26’ 8”. 

The one being in north lat., the other in south, to find the diff. of lat., add. 

30. A grocer, having on hand 17 cwt. 3 qr. 5b. of sugar, buys 
5 cwt. 20 Ib. more, and then sells 12 cwt. 1 qr. 51b. 80z. How 
much has he remaining ? Ans. 10 cwt. 2 qr. 19 Ib. 8 oz. 

31. From a piece of cloth containing 87 yd. 1 nail, are cut 6 yd. 
3qr. 2nails, and afterwards 8yd. 8qr. 2na. lin. How much is 
then left ? Ans. 21 yd. 1 qr. 1fin. 

32. Napoleon was born Aug. 15, 1769; Wellington, May 1, 
1769. Which was the older, and how much ? 

33. How old was Napoleon when the battle of Waterloo took 
place, June 18, 1815? How old was Wellington ? 

34. A druggist, having bought 1lb. 8% ofsalts, put 43 33 15 
in one bottle, and 8 3 25 19gr. in another. How much did what 
was left weigh ? Ans. 73 33 25 gr. 

85. From 11b. Troy take 10 oz. 17 pwt. 18 gr. 


COMPOUND MULTIPLICATION, 175 


CHUA PTE. R XN, d. 


COMPOUND MULTIPLICATION. 


295. Compound Multiplication is the process of taking 
a compound number a certain number of times. It com- 
bines multiplication and reduction ascending. 

ExampLE.—Multiply 4 gal. 2 qt. 1 pt. 3 gi. by 36. 

Write the multiplier under the lowest de- 


nomination of the multiplicand. Begin to gals’ gta] pta ten 
multiply at the right. Age Oe a 8 
3 gi. x 36 = 108 gi. — 27 pt. Write 0 36 


under the gills, and add 27 pt. to the next See ees 
product. I pt. x 86 = 36 pt., and 27 pt. are 478-169 3 1 O 
63 pt. = 31 qt. 1 pt. Write 1 in the column : 
of pints, and add 381 qt. to the next product. 2 qt. x 86 = 72 qt., and 
31 qt. are 103 qt. = 25 gal. 8 qt. Write 3 under the quarts, and add 
25 to the next product. 4 gal. x 86 = 144 gal., and 25 gal. are 169 gal. 
Ans. 169 gal. 3 qt. 1 pt. 


In stead of multiplying pal. qt. pt. gi gal. qt. pt. gt 
by 86 at once, we may YS) te CD Ao - Dien fon rey 
multiply in turn by any 4. 6 
factors that will produce Syrmar: 

386; as, 4 and 9, or 6 18 Bey bs 0 PBA phi) a 2 
and 6. The product will _ é 9 ‘ 6 
be the same. Ue Oot su) OOF ot the mee 


296. Rurze.—1. Write the multiplier under the lowest 
denomination of the multiplicand. 

2. Beginning at the right, multiply each denomination 
in turn, and write the product under the number multi- 
plied, uniess it can be reduced to a higher denomination. 
In that case, divide ti by the number that it takes to make 
one of that denomination ; write the remainder under the 
number multiplied, and carry the quotient to the next 
product. 


2597, When Compound Division has been learned, Compound Multi- 
plication is best proved by dividing the product by the multiplier, and 
seeing whether the multiplicand results. 


295. What is Compound Multiplication? What processes does it combine? Go 
through the given example, explaining the steps,—296. Recite the rule.—297. How 
is Compound Multiplieation best proved? 


-/ 2 ae 


176 COMPOUND MULTIPLICATION. 


298, Multiply by 12 or less in one line. If the multiplier exceeds 
12 and is a composite number, it may be best to multiply by its factors. 
In this case, to prove the result, multiply by the factors in reverse order. 


999, If a fraction occurs in the product, it must be reduced to lower 
denominations, if there are any, and the result added in. See Examplee 
1 and 2 below. 


EXAMPLES FOR PRAOTICE. 


@) (2) 

nd.- “yd. ft. 1m! A. R&R. sq.rd. sq. yd. sq. ft. sq. in, 
Multiply = 3 Ino ly 6 Beret neon OB 
By 5 Wa 
16° 14/9 4 4 B os Ps 400 A118] 6 

tya. = 144 £8q.yd.= 6 108 

Ans. 1640527 0.64) ty dae Le 8 ee 
(3) ‘ (4) (9) 

gale qt. “pt.” “gi. ewt. qr lb.~ oz. ‘dr. da. h. min. sec. 

PAS Sy) jets Rea) 16 1 28 14 #15 2.19. 47 58 

10 11 12 


6. Multiply 2° 13’ 12” by 45. _ 

7. Multiply 12 ch. 151. 5.7 in. by 55. 

8. Multiply £14 17s. 8d. 8 far. by 56. 

9. Multiply 3 sq. mi. 2R. 154 P. by 60. 

10. Multiply 5 Ib. 80z. 13 pwt. 19 gr. by 63. 

11. Multiply 5 Cd. 3 ed. ft. 8 cu. ft. by 72. 

12. Multiply 22 gal. 1 qt. 1 pt. 24.¢i. by 77. 

13. Multiply 5 T. 14 cwt. 20Ib. 6 oz. 15 dr. by 99. 

14. Multiply 12 Ib. 6dr. 2 sc. 18 gr. by 108 (9 x 12). 

15. Multiply 1 mi. 87rd. 4 yd. 2 ft. 9in. by 182 (11 x 12). 

16. A has 3 packages of silver, each weighing 1 lb. 1loz. 14 
pwt. 9gr. B has 7 packages containing 8 oz. 23 gr. each. Which 
has the most, and how much? Ans. A 11b. 20z. 16pwt. 10gr. 

17. From a pipe of wine holding 127 gal. 1 pt. were filled 25 

298. How should we multiply by 12 or less? If the multiplier exceeds 12 and isa 


composite number, how may it be best to proceed? In this case, how can we prove 
the result ?—299, If a fraction occurs in the product, what must be done with it? 


EXAMPLES FOR PRACTICE. 177 


demijohns, each containing 5 gal. 14gi. Hew much wine re- 
mained in the pipe ? Ans, 3 qt. 1 pt. 24 gi. 

18. D, having given C a note dated Aug. 2, 1864, and paid it 
Jan. 1, 1865, borrowed from C the same amount for a period 
three times as long. How long was that? Ans. 1 yr. 2mo. 27 da. 

19, If 5 suits, each requiring 6 yd. 1qr. 1na., are cut from a 
piece of cloth containing 40 yd. 3 na., how much will remain ? 

20. Henry Smith bought of Walter Rowe, of Liverpool, 2 bar- 
rels of flour, at £2 4s. 6d. per bar.; 27 1b. coffee, at 114d. per Ib.; 
14 boxes sardines, at 8s. 6d. a box; 210]b. citron, at 1s. 8$d. Paid 
on account £3 17s. 64d, Make out Smith’s bill, showing the bal- 
ance due. Ans. £22 84d. 

21. A printer, having on hand 4 bundles of paper, printed 
three pamphlets, each requiring 1 ream 6 quires 12 sheets. How 
much paper had he then left ? Ans. 2 bundles 12 sheets. 

22. Ifto a pile containing 20 Cd. 8 ed. ft. of wood, 18 loads of 
1 Cd. 15 cu. ft. each, are carted, how much wood will there then 
be in the pile? Ans. 35 Cd. 4cd. ft. 3 cu. ft. 

23. A lady, having subscribed 100 guineas for the poor, pays 
four instalments of £15 8s. 64d. each. How much has she yet to 
pay ? Ans. £48 5s, 11d. 

24. P and Q start from two points 175 miles apart, and walk 
towards each other. P averages 15 mi. 20rd. 4yd. a day, and Q 
12 mi. 1 fur. 2yd. 2ft. How far apart are they at the end of five 
days ? Ans. 39 mi. 18rd. 5 yd. 6 in. 

25. If six farms, each containing 40 A. 2h. 15P., are taken 
from a section of land, how much remains? Ans. 396 A. 1R. 30P. 

26. Multiply 5 bu. 3pk. 6 qt. 1pt. by 7; 18; 23; 17; and add 
the products. Ans. 357 bu. 6 qt. 

27. Multiply 10 cu. yd. 19 cu. ft: 1123 cu. in: by 11; 19; 29; 41; 
and add the products. Ans. 1072 cu. yd. 20cu. ft. 1708 cu. in. 

28. Multiply the sum of 40ch. 991. 3.92in. and 89 ch. 4 in. 
by 50. 

29. From a heap of potatoes containing 243 bu. 2pk. were 
filled 150 baskets, each holding 3pk. 2qt. How many bushels, 
&c., of potatoes remained in the heap ? 


178 COMPOUND MULTIPLICATION, 


3000. DirrrerENcE or Time anp Loneirupr.—All 
places have not the same time. When it is noon here, 
it is sunset at some place east of us, and sunrise at some 
place west. 

This is because the carth turns on its axis from west to east. Places 
east of a given point are, therefore, brought within sight of the sun be- 
fore that point is, and have the sun in their meridian sooner. 

301..The difference of time between any two places 
being known, their difference of longitude can be found. 
The earth turns on its axis once in 24 hours. <A given 
point on its surface, therefore, completes a circle of 360° 
in 24 hours, moving 15° in 1 hour, 15’ in 1 minute, 15” 
in 1 second. Hence, 

To find the difference of iat in degrees, minutes, 
and seconds, multiply the difference of time, expressed in 
hours, minutes, and seconds, by 15. 


Navigators thus determine their longitude at sea. Taking with them 
a chronometer (an accurate watch) set to mark the time at a given place 
(as, Greenwich or Washington), they ascertain by an astronomical obser- 
vation the time at the spot they are in, reduce the difference of time to 
difference of longitude by the above rule, and thus find that they are so 
many degrees east or west of the meridian of the place for which their 
chronometer is set. 


Ex.— When it is noon at San Francisco, it is 4 min. 52 
sec. after 3 p.m. at Philadelphia. What is their differ- 
ence of longitude ? | 


h, min. sec. 
Difference of time, 3 4 52 
15 


Difference of longitude, 46° 13’ 0” 


30. The difference of time between Washington and Dublin 
is 4h. 42min. 51sec. What is their difference of longitude ? 
Ans. 70° 42’ 45”, 
31. When it is midnight at Detroit, it is 41 min. 18 sec. after 
5 s.M. at Paris; what isthe difference of long.? Ans. 85° 18’ 15”. 


300. What is said of the difference of time at different places? Why is this ?— 
801. Give the rule for finding the difference of longitude, when the difference of 
time is known. Why do we have to multiply by 15? How do navigators deter- 
mine their longitude at sea ? 


COMPOUND DIVISKON, 179 


CHAPTER XVII. 
COMPOUND DIVISION. 


302. Compound Division is the process of dividing a 
compound by an abstract number, or finding how many 
times one compound number is contained in another, 
It combines division and reduction descending. 

Ex. 1.—Find sy of 32rd. 4 yd. 3 ft. 


The divisor being greater than 12, 


we must use Long Division. Write the seg ty 
divisor at the left of the dividend, and 2D ouee Lea o Chris 
begin to divide at the left. 29 

Divide 32rd. by 29: quotient, 1rd. ; 8rd. 
remainder, 3rd. To continue the divis- BL 
ion, reduce the remainder to yards, and 
addin the 4 yd. in the dividend. 8x54 29) 205 203 yd. (yd. 
= 163. 1644+4= 20}. | 

99 is not contained in 204; hence 29) 644 ft. (2 ft. 
we have Oyd. for the quotient. Re- 58 


duce 20}yd. to feet, and add in the 


3 ft. of the dividend. 20} x 3— 61}. 6} ft. : 
61443 = 64}. 12 

Divide 644 ft. by 29: quotient, 2 ft. ; 29) 78in. (2 $9 in. 
remainder, 64 “ft. Reduce the remainder 58 
to inches, afd again divide. 64x12 “90 
= 78, 78 +29 — 23-9 in. Collect the z ee 
several quotients for the answer. Ans. 1rd. 2ft, 25§ in. 


Kix. 2.—How many powders weighing 15 5gr. each 
can be put up from a mixture containing 13 43 145°? 
As many as 15 5er. is contained times in 1% 43 155. Reduce 


both divisor and dividend to grains, that being tho lagrest denomination 
in either, and then divide. 


TS ber, == 25 er. 
1% 43 145 = 750¢r. 
750 er. + 25 gr. = 80 Ans. 


803, Rute.—1. Zo divide a compound by an abstract 
number, beginning at the left, divide each denomination 
in turn. When there is a remainder, reduce it to the next 


302. What is Compound Division? What processes does it combine? Ge 
through Examples 1 and 2, explaining the steps.—303, Recite the rule, 


180 COMPOUND DIVISION, 


lower denomination, add in the number of that denomi: 

nation in the dividend, if any, and continue the division. 
Collect the several quotients, each of the same denomina- 
tion as its dividend, for the entire quotient. 

2. Lo divide one compound number by another, re- 
duce both to the lowest denomination in either, and divide 
as in simple numbers. 

3. Prove by finding whether the product of divisor and 
quotient equals dividend. 


Divide by 12 or less in one line. If the divisor exceeds 12 and is a 
composite number, its factors may be used in dividing. 


EXAMPLES FOR PRACTICE. 


(1) (2) 
eeCyyite dts Lbs an OF. OL: Bq.1mi.° AL. RR. sq. rd. ‘8q.{yd. 
giv 98 TQ) TLS spell) Dina as 


3 
2 FiO rpbO 82 “5 Ans. 9 874.105 1532 


3. Divide 22.sq. yd. 6 sq. ft. 85 sq. in. by 11. Same ans. 

4. Divide 47 sq. yd. 4sq. ft. 1124 sq. in. by 23. | Jor both. 

5. Divide 20 yd. I qr. Ina. by 9. Ans. 2yd. 1 qr. }na. 

6. Divide 228 ch. 391. 4.62in. by 57. Ans. 4ch. 5.5 in, 

7, Divide 3cu. yd. 20 cu. ft. 709 cu.in. by 401. Ans. 487 cu. in. 

8. Divide 1 yr. 27 da. 22h. 80 min. 30sec. by 65. 

9. Divide 127 lb. 10 0z. 18 pwt. 19 gr. by 164. 

10. Divide 48 Cd. 4cd. ft. 11 cu. ft. by 19. 

11. Divide 101 chal. 34 bu. 63} pk. by 83. 

12. Divide 6 mi. 81rd. 4 yd. 2 ft. 2in. by 42 (7 x 6). 

18. Divide 12 1b 11% 73 25 19gr. by 121 (11 x 11). 

14. Divide £147 17s. 4d. 2far. by 8; by 18; by 29; and add 

the quotients. And £69, 12s. 0d. 325 far. 

~ -'15. Divide 57 cwt. 15 1b. 50z. by 8; by 18; by 58; and add 

the quotients. Ans. 12 cwt. 2 qr. 2 Ib. 9oz. 7335 dr 
16. Divide 8 fur, 4 yd. 6in. by 34; by 14; find the Micaee 

between the quotients. ae 5rd, 1ft. 27445 in, 
17. Find 3, of 18 T. 14 cwt. 3 qr. 15 Ib. 


EXAMPLES FOR PRACTICE. — 181 


18. Find? of 20bu. 8pk. 7 gt. lpt. Ans. 11 bu. 2pk. 5 qt. 4 pt. 


Multiply the compound number by the numerator of the fraction, and divide 
the product by its denominator. 


19, Find 7, of 3 tb 18 gr. Ans. 1ib 23 63 25 243 gr, 
20. Find 43 of 7 guin. 10s. 6d. Ans. 6 guin. 20s. 3d. 
21. How many times is 2 cu. ft. 84cu. in. contained in 1 cu. yd. 
5 cu. ft.? (See Example 2, p. 179.) Ans. 154473 times. 
22. How many spoons, weighing 1 oz. 9 pwt. 13 gr. apiece, can 
be made out of 1 lb. 18 pwt. of silver ? Ans, 832%, 
23. D, having 431A. 3R. 21P. of land, bought 126A. 31P. 
more, and then divided the whole equally among his 4 sons and 
8 daughters. How much did each receive? Ans. 79 A. 2R, 36P. 
24, From a puncheon of rum, containing 80 gal. 1 qt. 14 pt., 
2 qt. leaked out, and what remained was put up in bottles holding 
1 pt. 1gi. apiece. How many bottles were tilled ? Ans, 5113. 
25. A lady went out with £20, and spent £3 6s. 3d. How 
many books, at 3s. 83d., could she buy with what remained ? 
26. What is the average speed per minute of a train that runs 
*30mi. 30rd. 5yd. in one hour, and 27 mi. 4fur. 80rd. 1 yd. the 
next ? Ans. 3 fur. 33rd. 4 yd. 1 ft. 104 in. 


How far did the train go in two hours? How many niinutes in 2 h.? The 
average rate per minute will be ;3, of the distance travelled in 2 h. 


27. B has as much silver plate as his father, who has 18 Ib. 
4o0z. At 5c. an ounce, what tax has B to pay on this silver, 40 oz. 
being exempt from taxation ? 

28. How many times longer is a field 13 ch. 231. 1.4%in. in 
length, than one that measures 1ch. 11, 64in.? © Ans. 18 times. 

29. Two fields, of 14 A. each, produce respectively 36 bu. 8 pk. 
and 34 bu. 2pk. 1 qt. of wheat. What is the average yield per 
square rod? Ans. 443, qt. 

30. During February, 1864, a grocer sold 15 ewt. 20Ib. 8 oz. 
of sugar; what was his average daily sale? Ans. 52 1b. 6302, 

31. A druggist, having 3ib 8% 25 of soda, put up from it 3 
dozen powders of 143 each, and divided the rest into 6 equal 
parts; what did each of these weigh? Ans. 6% 23 2D 62er. 

82. A person owning a section of land sold 50 A. 1R. 22sq. 


182 COMPOUND DIVISION. 


rd., and gave away 20 A. 39sq.rd. 80sq. yd. What remained, he 
divided equally among his five sons. What was each son’s share? 
Ans. 113 A. 3R. 19 sq. rd, 18} sq. yd. 

33. Twenty-four men agree to construct 7mi. 1 fur. 24rd. of 
road; after completing 4 of it, they employ 8 more men. What 
distance does each man construct before and after the 8 men were 
employed ? Ans. 16rd. before; 1 fur. 20 rd. after. 


304, DirrerENcE oF LonerrupE anp Trve.—1 hour 
being the difference of time for 15° of longitude (§ 301), 
1 minute for 15’, 1 second for 15”, 

To find the difference of time between two places, in 
hours, minutes, and seconds, divide their difference of 
longitude, in degrees, minutes, and seconds, by 15. 

When the time at a given place is known, add the dif- 
JFerence to jind the time of an y place east of at, subtract 
for any place west. 


Exampie.—sSt. Petersburg is in 30° 19’ K., New York 
in 74° 3” W. longitude. When it is 3p.m. at N. Y., 
what o’clock is it at St. Petersburg ? 


Difference of longitude, 104° 19’ 3” 
104° 19' 38" +15 = 6h. 57min. 16sec. + Diff. of time. 
St. Petersburg being east of N. Y., add: 3h. + 6h. 57 min. 16 sec. 
Ans. 57min. 16sec. past 9 P.M: 


34. When it is noon at Buffalo, what is the time at Naples, 
the former being in 78° 55’ West longitude, the latter in 14° 15’ 
Last ? Ans. 12 min. 40sec. past 6 P.M. 

35. When it is 6 o’clock a.m. at Portland, what is the time at 
San Francisco, the former being in 70° 15’ W. long., and the lat- 
ter in 122° 23’ W.? Ans. 31 min. 28 sec. past 2 A. M. 

36. Required the difference of time between Buffalo and San 
Francisco. Ans. 2h. 53 min. 52 sec. 

37. Between Portland and Naples. Ans. 5h. 38 min. 

304. How may the difference of time between two places be found, when their 


difference of longitude is known? In what case must the difference of time be 
added, and in what caso subtracted ? 


MISCELLANEOUS: EXAMPLES. 183 


305, Miscerzanrous Exampies In Compounp Numbers. 


1. If a man wastes 4 minutes a day, how much time will he 
waste in the years 1867, 1868 ? Ans. 2da. 44min. 

2. A druggist bought 3lb. 140z. Av. of magnesia; he sold 5 
packages of 1 dr. lsc. each; how many pounds, &c., Troy: has he 
remaining? (See § 239.) Ans. 3lb. 80z. 5pwt. 164 er. 

3. What is the difference of cost between 8 tons of hewn 
timber, at $1 a cu. ft., and 24 tons round timber, at 88c. a cu. ft.? 

4, What fraction of 1 mile is 6 fathoms ? 

5. A grocer’s quart measure was too small by half a gill. How 
much did he thus dishonestly make in selling four barrels of cider, 
averaging 34 gal. 2qt. 1 pt. each, if the cider was worth 24 cents a 
gallon ? Ans, $2.216. 

6. A lady, for ten successive years, went into the country on 
the 20th of May, and returned the 17th of the following October. 
At 90c. a day, what did her board cost her for the whole time? 
(See Table, § 271.) Ans. $1850. 

7. A farmer owns a horse 15 hands high and a lamb 1% ft. 
high. What common fraction, and what decimal, is the lamb’s 
height of the horse’s, and how much higher is the horse than the 
lamb ? Ans. $; 3; 8ft. 4in, 

8. Which is the greater, .65 1b. Troy or 472 Ib. yt ? 

9. Washington was born Feb. 22, 1132: died Dec. 14, 1799. 
Franklin was born Jan. 17,1706; died April 17, 1790. How much 
did Franklin’s age exceed Washington’s? Ans. 16yr. 5mo. 8da. | 

10. From a hogshead containing 63 gal. of wine, 1 pt. leaked 
out; what fraction of the original quantity was thus lost? Ans. x47. 

11. From 1qt. 1 pt. of grain was raised 1 bu. What decimal 
was the seed of the crop ? Ans. .046875. 

12. How many angles of 3° 45’ will fill the same space as 1 
right angle, of 90° ? 

18. How many square yards of carncuam will be required for 
a room 26 feet by 32 feet ? 

14, What part of 1 perch is ;2; of an acre? Ans, 272P. 


184 ADDITION AND SUBTRACTION 


306. Zo add or subtract denominate fractions, com 
mon or decimal, of different denominations. 


8. d. far 

Ex. 1.—Add £3, 3%s., and 4d. &3 Pe 7 ; : 

1. Reduce each fraction to integers of sh ae 9 
lower denominations (§ 281), and then add. SULLY fe ae 
Ans 6B solvate 

L3b et Bg wae U1. : 
rt i an = 138. 2. Or, reduce £8 to shillings, and add in 
Ty + yp = Babs. 68. Reserving 8, the integer, reduce the 
78 = $d. fraction +s. to pence, and add in 4d. Re 

#d. + 4d. = 14d. serving 1, the integer, reduce id. to farthings. 

Mines aiaeee Finally, collect the integers for the answer. 


Ex. 2.—From .825 T. subtract .62 ewt. 
Proceed by either method shown under Example 1. 


ewt. qr. Ib. 825 T.'= 16.5 wt. 
cores Sen LOE Elba: ARN Signe NEDA AO eerie 
Pee SaaS 15 ewt. 3 qr. 131b. Ans. 
875 Ib 
Ex. 3,—Add .875 th, .7 3, and .4.3. 12 
In adding decimals of different denomina- 10.500 3 
tions, the second method is generally prefer- : iT 
able. 11) .2 3 
Reduce .875 th to ounces, and add in .73. 8 
Reserving 11, the integer, reduce .2% to drams, Se 
and add in .43%.. Collect the integers for the 63 
answer.) Ans. 11%.23. — uA 
2| .03 


Rutz.—1. Leduce the gwen fractions to integers of 
lower denominations , then add or subtract, as required. 

2. Or, reduce the fraction of the highest denomination 
to integers of lower denominations, taking care, as each 
ts reached, to add or subtract, as may be required, any 
given fractional term belonging to that denomination. 


15. Add }cwt., $qr., and 1b. Ans, 2 qr. 8lb. 9 oz. 54 dr, 
16. Add ;, bu., ¢pk., and i qt. Ans. 55); qt 


806. In how many ways may we add or subtract fractions of different denomina- 
tions? Jllustrate these two modes with the given examples. Recite the rule, 


OF DENOMINATE FRACTIONS. 185 


17. From 3 oz. take 3 pwt. Ans, Tpwt. 15 gr. 
18. From .375 da. take .2 min. Ans. 8h. 59 min. 48 sec. 
19. From .22 ch. take .431. Ans. 211. 4.5144 in. © 
20. Add £.75, .8s., .36d., .9 far. Ans, 15s. 10d. 0:74 far. 
21. Add 2wk., }da., th. Ans. 4da. 21h. 8min. , 


22. Add 3mi., §fur., 4rd., yd., $ft., 254 in. Same ans. 
-23. Add 3mi., {fur., 128 rd., Syd., 2 ft., +3, in. Sor both. 
24, Add .6cu.yd., .875 cu. ft., 4cu.in. Ans. 17 ¢. ft. 1806. in. 
125. Add .3875sq.mi., .54A., .6R. Ans. 240A. 2R. 30.4sq.rd. 
26. From § A. take 4 ot 8 roods. Ans. 2R. 42 P. 
27. From thhd. take 2 qt. Ans. 6 gal. 3 qt. 3 pt. 
28. From .32 ib take .9%. ‘Ansu2 on, VT dy. G1.865 112 er. 
29. From 4 of 3 of a day take 4 of 11 hours. 
30. Add 1Ib. Troy, 40z., and 4pwt. Ans. 2.0z. 13 pwt. 32 er. 
31. A man had to plough 8A.,4R.,$P. When 3A. 2R. 2P. 
was ploughed, how much had he todo? Ans. 2A. 1R. 10+; P. 
32. How many cu. in. in 8 gal. 2qt. 1pt., Wine? Ans. 8373. 
33. In 3 ofa gallon + 1 of a quart, Beer? Ans. 223.25 cu. in. 
34. In 1 bushel 8 pecks ? Ans. 3763.235 cu. in. 
35. How many feet in 3 of a chain + ,8,fur.? Ans. 2473 ft. 
36. From a piece of cloth containing 20 yd. 2 qr. 2 nails, 8 suits, 
each requiring 44 yd., were cut. One third of the remainder was 
cold for $10.682; what did it bring per yard ? Ans. $4.50. 
3 remainder = 2yd, lqr. 2na.= 2,375 yd. $10.6875 + 2.3875 = $4.50. 
37. What cost 4bu. 3 pk. 6 qt. of potatoes, at 75c. a bushel? 
As the price is given by the bushel, reduce, by § 285, 4bu. 3pk. 6qt. to bushels 
end the decimal of a bushel (4.9375 bu.), and multiply by the price. Ans. $3.70. 
38. What cost 4 bu. 3 pk. 6qt. of potatoes, at 18c. a pk. ? 


As the price is given by the peck, reduce 4bu. 3 pk. 6qt. to pecks and the 
decimal of a peck, and multiply by the price. 4bu. 3pk.=19pk. 6qt.= .75pk. 
19.75 x .18 = $3,555, Ans. 


39. Whatcost 57A. 2R. 20P., at $20 anacre? Ans. $1152.50. 


At $3.75 a rood ? ; Ans. $864.3875. 
40. What cost 7 gal. 3 qt. 1 pt. of wine, at $8 a gal.? Ans. $63. 
At $1.50 a quart ? Ans. $47.25. 


41, What cost 5T. 17 cwt. 20 Ib. of hay, at $30.50 a ton? 
At $1.60 a hundred-weight ? Ans. $187.52, 


186 PRACTICE. 


42, What cost 3lb. 6 oz. 1dr. 2sc. of quinine, at $2.75 per oz. ? 
43. Find the cost of a gold ornament, weighing 40z. 18 pwt. 


20 gr., at £4 9s. per ounce. Ans. £21 19s, 9d. 2.8 far. 
44, What is the cost of a block of marble, 9 ft. long, 4ft. 4 in. 
wide, and 3 ft. Gin. thick, at $5 a cubic foot? Ans. $682.50. 


45. What cost 8 bundles 8 quires of paper, at $6 a ream ? 

46. What cost a field 4 ch. 801. square, at $6.25 a rood ? 

47. What cost 44 A. 8} R. 4P. of land, at $4.25 a rood? 

48. What cost }T. } ewt. 251b., at $3 per cwt.? Ans. $11.35. 
49. 18 Cd. 8 cu. ft. of wood, at $8 a cord ? Ans. $144.50. 
50. 7T. 14cewt. 3 qr. 101b., at $75 aton? Ans. $580.687 +. 
51. 3ib.63 15 gr. of calomel, at $1.50 an ounce ? 


Practice. 


307, Practice is a short method of operating with com- 
pound numbers, by means of aliquot parts. It was ap- 
plied to Federal Money on p. 123, and may be extended 
to compound numbers generally. The aliquot parts most 
frequently used are as follows :— 


Taste oF Axiguor Parts. 


Sterling Money. Avoir. Wt. Time. 


s. d Eon ede s. | Ib. cwt.| mo. da. yr.|da. mo. 
Ace cis) abi sO cerned D0 aronmeeak af ey tal ie we 
6 474° £) 88h = FY 4 =+4+/,)10 = 4 
5 BSS SE RIB aes eas 718 Se Ps GO 
4 ite 2 Se ek 20 eye ee eh DES eas Pm 
3 & | 1d = § | 163 = $$) 2 =%3| 3 = 
1 2 $. | Govier 124.0 44 bh Toe Ba aie 
2 to 10 = i leew 
1 12 Be. ay] td = qe 
1 20 5 = | 30 da. allowed to 1 mo. 


807. What is Practice? Give the aliquot parts of £1. Of a shilling. Of a hun- 
dred-weight. Of ayear. Of a month. 


PRACTICE. 187 


Ex, 52.—What cost 960 Grammars, at 1s. 8d. each? 


At £1 each, 960 Grammars would cost £960. 12) £960 
But Is. 8d. = an of £1; therefore, at 1s. 8d., they 7 ae Y-%), 
will cost 3b; of £960, or "£80. ns. 

Ex. 53.—If it costs $17.50 to insure a house 1 year, 
what will it cost to insure it for 8 yr. 1 mo. 15 da. ? 


1mo, 15 da. = 3 | $17.50 For 3 years take 3 times the cost 
for lyr. For 1 mo, 15 da., which is 
$52.50 4 of Lyr., take 4 of the cost for 1 yr. 


9.1875 Find the whole by adding these two 
Ans. $54.6875 Pans 


Ex, 54.—How much seed will be needed for 10 A. 1R. 
30P., allowing 1 bu. 2 pk. 4 qt. to an acre? 


bu. pk. qt 
For 10 A. take 10 times the quantity TR) 1 See 
required for 1 A. For 1R., which is } 10 
of 1 A., take $ the quantity required for ——— 
1A. 30 P. not being an aliquot part of | L0G Laan 
1 rood, take first for 20 P., which is 4 | Oe" Dee 
of 1R.; then for 10P., which is 4 of ZO teraas a Ope =, Ga 
20P, Find the whole by adding these 10P. 417 02 0. 28.25 
parts. Ans. 16 8 6.75 
55. What cost 14 dozen Readers, at 3s. 4d. apiece? Ans. £28. 
56. 14 ewt. 1241b. of cheese, at $16 a cwt.? Ans. $226. 
57. 1 gross of knives, at 2s. 6d. apiece ? Ans. £18. 


58. 5yd. 1qr. Ina. of cloth, at $6.25 ayd.? Ans. $33.20+. 

59. 26 gal. 1qt. 1 pt. 1 gi. of wine, at $7 agal.? Ans. $184.84+. 

60. What will it cost to travel 1200 miles, at 14d. a mile? 

61. At $25 a month, what will be a man’s wages for 1 year 
7 months 12 days? 

62. What will be the yield of 16 A. 25P. of land, at the rate 
of 24 bu. 3 pk. 1 qt. per acre? Ans. 400 bu. 1 pk. 3.90625 qt. 

63. What cost a plate of glass, measuring 7 ft. by 5 ft. 6 in., at 
4s. 6d. a square foot ? Ans. £8 18s. 3d. 

64. What cost 10 panes of glass, each 4ft. by 2 ft. 9in., at Is. 
8d. a square foot? Ans. £6 7s. 6d. 

65. Find the rent of a furth of 250 A. 2R. 15P., at £1 12s. 4d. 
an acre. Ans. £405 2s. 6d. 1.5 far, 


en" 


188 DUODECIMALS. 


GN Bis Sah ed hd rgd be phi 


DUODECIMALS. 


308. Duodecimals are a system of compound numbers, 
sometimes used as measures of length, surface, and solidity. 

The foot, whether linear, square, or cubic, is the unit ; 
and the other denominations arise from successive divis- 
ions by 12. Hence the term duodecimals, duodecim bein 
the Latin for twelve. 


1 of any denomination in this system makes 12 of the next lower; 
and, conversely, 12 of any denomination make 1 of the next higher. 


TABLE. 


1 foot (ft.) = 12 primes, marked’. 


1 prime = 12 seconds, marked”. 
1 second = 12 thirds, marked’”. 
1 third. = 12 fourths, marked’, &c. 
i wet —_ 12 rere 
Ast — 1B)" — 144." 
1 t — 12 ad — 144 ‘rr — 1728 my 
ere 0 1s SO eee (2. ee 


The marks used to distinguish the denominations 
(Qo ” '’) ave called In'dices (singular, Index). 


309, When used in connection with one dimension simply, as length 
or breadth, the prime, being +4; of a foot, is equivalent to 1 inch. 

When ‘applied to surfaces, the pe being +4; of a foot, equals 12 
square inches. The second, being +5 of 75 of a foot, equals 1 Sq. ie 

When applied to solid contents, the Eee being ls of a foot, equals 
144 cubic inches; the second = 12 cubic inches; the third = ‘| cubic 
inch. 


308. What are Duodecimals? What is the unit? How do the other denomina- 
tions arise? Whence is the term dwodecimals derived? Recite the Table. What 
are the marks used to distinguish the denominations called ?—309. What is the prime 
equivalent to, when used in connection with one dimension simply ? When applied 
to surfaces? When applied to solid contents?—310, How are duodecimals added, 
subtracted, multiplied, and divided ? 


DUODECIMALS. 189 


310. Duodecimals may be added, subtracted, multi- 
plied, and divided, like other compound numbers. 


EXAMPLES FOR PRAOTICE. 


(1) @) 
MOO titeeti OO 1 ore Multiply 6-8 oe 
ait, gt anibery ogi By 12 
Lt’ 8” g’”’ Silly lee 71 Ea PPE 
Of, 0! G6. gl gu Ans. 6 ft. 8-9" 0" 


ee 


YY TAs Pe les i a PE 


(4) 

(2 Divide 9ft. 1” 4’ by 82. 
From 8ft. 1’ 6” 6 10" AY SOG eee 
Take 5'ft. 0/14 9041" 8) 2ft. 3’ 07” 4/7 
Ans. fal 0's tk 8 Vt Ans. 0 T6213) 475.6" 2i6™ 


5, Find the sum of 3 ft. FP 5"", 16 ft. 6.7", 19 ft. 8°97" 11" 


PUI OCS BM and: b fee) ab Ans. 45 ft. 2° 7". 
6. From 25 ft. 1” take 16 ft. 8’ 9'" 8'"".., Ans. 8 ft. 9° 2° 4° 
7. Multiply 3 ft. 6’ 5” 7'” by 12. Ans. 42 ft, 5! 7”. 
8. Divide 6 ft. 4’” 10’”" by 81. Anse! 8" 10" Filta, 


9. From 100 ft. subtract 7 times 8’ 9’. Ans. 95 ft. 3’ 6” 9'". 
10. From 59 ft, take +t, of 6ft. 6”. Ans. 58 ft. 7" 5! 7!" 6” 
11.. Add-365-ft.-1/ 7” 9°" 8%") 521 ft." 10" 107 11’; 605, ft. 8.8" - 
1”, and:731 ft; 30.8". Ans. 22241. 3" 67, 
12. What is the sum of 14 ft. 5’ 6’” 9’ and 11’ 11” 10’ 107"? 

What is their difference ? : 
13. What is the sum, and what the difference, of 47 ft. 1’ 1” 
ear Lett 1 eT ee OF 10" 10" 10" “and 10 10 10e eg 
. 14. From the sum of 8’ 9’ 8’ and 10’ 10” 10” take the sum 


of 4-8. US endal ees 317", Ans. 1 ft. 2' 9". 
15. Multiply by 36 the sum of 8” 8’, 4’ 9”, and 2 ft. 3’ 4° 7", 
Divide the product by 7. : Ans. 14 ft. 9" 5/7 1" +. 


16. What is the sum of 100 ft. 8’ 8”, 185 ft..1" 9’, 65 ft. 9’ 2” 
i", AB ft. 3 8’’, and 200 ft, 6’ 6 8”? 

17. Which is greater, } of 10 ft. 6” 7” or 4 of 80 ft. 1° 1”, and 
how much? 


190 DUODECIMALS. 


$11. Muurrerication or Duoprecimats By Dvuopxct. 
MALS.—1 ft. is the unit. Hence, multiplying by 1 ft. is 
simply multiplying by 1, and the denomination of the 
product will be the same as that of the multiplicand. 
oy elite 3. 

1’= +A, ft. Hence, multiplying by 1’ is multiplying 
by 35, and the denomination of the product will be one 
degree lower than that of the multiplicand. 3' x 1'= 3", 

teh = 7 1. of 4, ft. Hence, multiplying by 1” is multiply- 
ing by +5 of +45, and the denomibation of the product 
will be ae degrees lower than that of the multiplicand. 
3! x 1% — gir 

1” = A, of +4 of {5 of a ft. Hence, multiplying by 
1’ is multiplying by +4 of 74 of 35, and the denomina- 
tion of the product will be three degrees lower than that 
of the multiplicand. 3’ x 1" = 3”, 

From the above it will be seen that The index of a prod- 
uct equals the sum of the indices of its factors. 

Thus 634° 3's=18": 6° x 3” = IB 6" HS Hah Se SF OS IB, 


ExampLe.—Multiply 14 ft. 7’ 8" by 2 ft. 6’. 


Set the multiplier under the multipli- 14ft 7 8” 
cand, with their right-hand terms in line. Early 


Begin to multiply at the right, reducing 2 ft. 6 
and carrying as in compound multiplica- Tite. a06 On mae 
tion. 29 ft. 3’ 4" 

8” x 6’ = 48"" = 4"; carry 4 to the Raste Le Pe 
next product. 7 x 6 = 42", and 4” ieee Bye 
carried makes 46” = 3’ 10”; write down 10”, and carry 3' to the next 


product. 14ft. x 6’ = 84’, and 3’ carried makes 87’ = 7 ft. 3’. 

Next multiply by 2ft., remembering that, when we multiply by feet, 
the product is of the same denomination as the multiplicand. Set the 
terms of this product under like en CE uve oun. in the former one. Final- 
ly, add the partial products. 


312, Ruru.—l. Write the multiplier under the multi- 
plicand, with their right-hand terms én line. 


811. How does the denomination of the product compare with that of the mul- 
tiplicand, when we multiply by 1ft.? When we multiply by 1’? When we multi- 
ply by 1”? When we multiply by 1/”? What rule is hence deduced, for the index 
of a product? Solve and explain the given example.—312. Regite the rule for the 
multiplication of duodecimals. 


DUODECIMALS. 19} 


2. Beginning at the right, multiply by each term of 
the multiplier, giving each product an index equal to the 
sum of the indices of tts factors, and reducing and carry- 
ing as in compound multiplication. Write terms of the 
same denomination in the partial products in the same 
column, and finally add the partial products. 


ERAMPRES FOR? PRAOTICE. 


i Multiphys fe72% by (£t.,6'.8o% Ans. 27 ft. (9% 6S 
. Multiply 7 ft. 8’ 9” by 6 ft. 4’ 3”; by 12ft. 5’; by 9 ft. 8”. 
. Multiply 6 ft. 9’ 7” by 4ft. 2’. Ans, 28 ft. 8’ 11" 2!” 
. What is the area of a slab 7 ft. 3’ long and 2 ft. 11’ broad ? 
. What is the area of a hall 37 ft. 3’ long by 10 ft. 7’ wide? 

. How many square ft., &c., ina garden 100 ft. 6’ by 89 ft. 7’? 
7. How many square ft. in 12 boards, each 12 ft. 8’ by 1 ft. 9’? 


Solve this and the next two examples by the above rule. Then prove the re 
sult by expressing the primes as fractions of a foot, multiplying, and reducing the 
fraction of a foot in the product, if there is any, to primes, &c. Thus, in Ex, 7:— 
12 ft. & = 122 ft 1ft. 9’ = If ft. 

‘ EN 6G Wien sq. ft. Ans. 


oO co FP OO ND 


8. How many cubic fect, primes, &c., in a wall, 80 ft. 9’ long, 
1 ft. 8’ wide, and 3 ft. 4’ high ? 

9. How many cubic feet in a pile of wood, 156 ft. long, 4 ft. 8’ 
high, 6 ft. 4° wide? How many cords? Ans. 862, Cd. 

10. A room is 18 ft. long, 14 ft. 6’ wide, 9 ft. 8’ high. It con- 
tains four windows, each 5 ft. 6’ by 3ft.; and two doors, each 
6 ft. 9’ by 2 ft. 10’. What will be the cost of plastering said room, 
at 25c. per square yard ? _ Ans. $21.81. 


The four walls and ceiling are to be plastered. 

Two of the walls have an area of 18ft. x 9 ft. 8’ each. 

The two other walls have an area of 14 ft. 6’ x 9ft. 8’ each. 

The ceiling has an area of 18 ft. x 14 ft. 6’. 

Deduct from the sum of these areas, the areas of the windows and doors, whick 
are not to be plastered: 4 windows, each 5ft. 6’ x 3ft.: 2 doors, each 6ft. 9' x 2 ft. 
10’. Reduce the sq. feet to sq. yards, and multiply by the price. 


11. What will it cost to paint a house 42 ft. 6’ deep, 28 ft. 6’ 
wide, and 19 ft. 6’ high, at 24c. per square yard, no allowance 
being made for windows? Ans. $73.84. 


. 


192 DUODECIMALS. 


313. Diviston or DuoprcimMaLts BY DvoDECIMALS. 


In multiplying duodecimals, we assign to a product 
an index equal to the sum of zy indices of its factors. 
Hence, in dividing, To find the index of the quotient, we 
subtract the index of the divisor from that of the dividend. 

Thus, 18” +6’ = $'; 18!” 6 = 8": 18’ = 6” — 8, 

314. If the index of the divisor exceeds that of the 
dividend, reduce the dividend to the same denomination 
as the divisor, and the quotient will be feet. 

Exampir.—Divide 18 square feet by 6”, 

18 ft. = 2592" 2592" + 6” = 432 ft, Ans, 

Exampie.—Divide 27 sq. ft..77 9'" 6"" by 3 ft. 7’ 2”. 


Write the divisor at the left of the dividend,“as in other cases of com- 
pound division. Begin to divide at the left. 


oF sq. ft. + 3 ft. P Y or ? " yn r of 
= 9ft. But, making 3ft. 7) 2") 27 ft. 0" 7” 9'" 6" CTH 6" 8" 


allowance for the 25 ft. 2! 2" NOE Otee IE Beh Ans. 
primes in the divi- Lip lO as ore 

sor, which amount Lite Oe 

to more than half 9 9 gt 

a foot, we write ait; 10” g!’ g/” 


as the first term in 
the quotient. We 
then multiply the whole divisor by 7 {t., and subtract the product from 
the dividend. 

3 fi. is not contained in 1 ft., the first term of the new dividend ; hence 
we reduce 1 ft. to primes, and add in 10’. Dividing 22’ by 3 ft. (making 
allowance, as above), we get 6’. Write 6’ in the quotient, multiply the 
divisor by it, and subtract. 

Dividing 10” by 8 ft., we get 3”, which we write as the third term in 
the quotient, Multiplying the divisor by this term and subtracting the 
product, we find there is no remainder. 


If, on multiplying the divisor by any term of the quotient, the product 
is greater than the partial dividend, the quotient term must be diminished, 


015. RutE—1. Divide the highest term of the divi- 
dend by that of the divisor, making tt divisible, if neces- 


813. In dividing duodecimals, how do we find the index of the quotient ?—314. 
{f the index of the divisor exceeds that of the dividend, how must we proceed? 
Solve and explain the given example. In what case must the term placed in the 
quoticnt be diminished )—815. Recite the rule for the division of duodecimals, 


is 


DUODECIMALS. 193 


sary, by reducing it to a lower denomination, and adding 
in the given number of that denomination. Write the 
result in the quotient, multiply the whole divisor by ut, and 
subtract the product from the dividend. 

2. Divide the highest term of the new dividend as 
before. Write the result in the quotient, multiply the 
divisor by it, and subtract. Proceed thus till the division 
terminates, or a quotient sufficiently exact is obtained. 


EXAMPLES FOR PRAOTIOE. 


1. Divide 32 ft. 9’ 9” by 7 ft. 3’ 6”. Ans. 4 ft. 6’. 
2. Divide 18 ft. 4’ 6’ by 3’ 6” (§ 314). Ans. 63 ft. 
3. Divide 82 ft. 9’ 9” by 29 ft. 2’, Ans, 1ft- 146% 
4, Divide 42 ft. 10’ 10” 4’” by 6 ft. 1’ 4”. Ans. 7 ft. 3”. 
5. Divide 9’ 11” 8'” 6’ by 47.3". Ans. 28 ft. 2’. 
6. What is the breadth of a marble slab, whose area is 21 ft. 
1’ 9”, and its length 7 ft. 3’? Ans. 2ft. 11’. 


7. A carpenter bought 920 sq. ft. of boards. If their united 
length was 480 ft., what was their average breadth ? 

8. A board fence 6 ft. 4’ high contains 510 ft. 10’ 8” of surface. 
How long is the fence ? | Ans. 80 ft. 8’. 

9. In digging a cellar 42 ft. 10’ long and 12 ft. 6’ wide, 4283 
cu. ft. 4’ of earth was thrown out. What was its depth ? 


Divide the solid contents, represented by the amount of earth thrown out, by 
the product of the two given dimensions. Ans. 8 ft. 


MIscELLANEOUS QuEsTIONS..-When do we add, to find the difference 
of latitude between two places? To find the difference of longitude? 
How is the difference of time found from the difference of longitude? 
How is the difference of longitude found from the difference of time? 

What is the unit of duodecimals? What is meant by the indices of 
uodecimals? What is the index of the foot? Of the prime? How 
many inches are equal to a prime, when used in connection with length 
or width only? When used in connection with surface? With solid 
contents? Recite the rule for multiplying duodecimals by duodecimals, 
How may the operation be proved? Recite the rule for dividing duo- 
decimals by duodecimals. How may the operation be proved? 


194 PERCENTAGE, 


CHAPTER XIX. 
PERCENTAGE. 


- 316. Per cent., from the Latin words per centum, means 
by or on the hundred. One per cent. means one on every 
hundred, or one hundredth ; it is written briefly 1%, and 
is equivalent to 54, or .01. Two per cent., 2 on 100, or 
two hundredths, is written 2%, and equals =2, or .02. 

317. Any per cent. or number of hundredths may thus « 
be written either as a common fraction or a decimal; but 
the decimal form is preferred, as easier to operate with. 


Any integral per cent. less than 100 is expressed by two decimal fig- 
ures. 21:4 == -012* 10-4 = 10. 

100 Z, being +83, is written 1.; 15047 =— 1.50; 200% = 2., &e. 

Any part of 1% may be expressed by taking the like part of .01: 

4% =40f Ol = .005. £4 = $ of .01 = .00875. 

Any part of 14 that can not be exactly expressed as a decimal may 
be written as a common fraction after the place of hundredths. Thus, 
4% = .004.' 4% = .008. 


318. The following examples will show how to ex- 
press different rates per cent. decimally :— 


1% = 07 | 525% = 5.25 4% = .00125 
124 = 12] 44 = .005| 44% = .041 

40% = 40] 44 = .00$ | 153% = .1575 
100% = 100; 44 = .002 | 232% = .234 


300% = 3.00| 4% = .004 | 304% = .807 


In the case of an integral per cent., the decimal point must not be 
prefixed when the sign % or the words per cent. are used. 25 is very 
different from .25%; the former being equivalent to 3375 or +,—the latter 
to You of Tho, OF zbo- 


516. What is the expression per cent. derived from? What does it mean?, 
What does one per cent. mean? How is it written? To what is it equivalent? 
Two per cent. ?—317, How may any per cent. be written? Which form is preferred, 
and why ? How many decimal figures are required to express any integral per cent. 
less than 100? How is 100 per cent. written decimally ? 150 per cent.? 200 per 
cent.? How may any part of 1 percent. be expressed? How may any part of t 
per cent, that can not be exactly expressed as a decimal be written ?—318. Give ex- 
amples of the mode of expressing different rates. What caution is given in the case 
of an integral per cent. ? 


.. Pon per 7 ; , 
: ‘ | 


EXERCISE IN PERCENTAGE, 195 


7. 


319. ExEROISE. 


1. Write the following rates per cent. as decimals: 64%; 444; 
2 3 1016s BBs 1BSs Es BOOK; Hs Bios SEK; 8085 

02%; 425 % 3 es 55 3 5 9% 5 Tz 0; 814% 5 153 $%3 985 % 5 
sete 584%; z45%; 100%; 10004. 

2. Read the following as so many per cent.: .0825 (eight and 
a quarter per cent.); .04; 2.00 (two hundred %); 17; .105; .20; 
4.00; .1175; .8384; 3.334; .034; .052; .052; .074 (72%); .094; 
1.15; .008; 8.00; .00$; .0003 (three hundredths of 1%); .0007. 

3. What per cent. is each of the following common fractions 
equivalent to? 4 (Annex two naughts to the numerator, and 
divide by the denominator: 1.00 + 2 = .50 = 50%); 4; 4; 4; 
$543 $5 93 tos tes 25 $5 85 3 85 G5 tos tes dos ass oe 

4. What common fraction is each of the following equivalent 
to? 25% (= fo = 2)5 4% (= 4 Of aby = zbo)s 7%; 64; 
14%; 20%; 10%; 40%; 504; 12%;4%; 8%; &%3 42%; 9% ; 
163%; 154; 60% 5 18%; 3 63 $%; 28%; 200%; £% 


320, In connection with the subject of Percentage, 
three things are to be considered :— 

1, The Rate, or number of hundredths taken. 
_ 2. The Base, or number of which the hundredths are 
taken. 

3. The Percentage, or number obtained by taking cer- 
tain hundredths of the base. 

Two of these being known, the third can be found ; 
for the Percentage is the product of the Base and Rate. 


Exameie 1.—How much is 7% of $16.85 ? 


Here the base and rate are given, and the per- 
centage is required. 7% is zi>. Taking jz is ? $16.85 
equivalent to multiplying by ;75 ($161). Hence we 07 
multiply the base, $16.85, by .07 (the rate expressed ae 
decimally), and point off the product as in Dike Ans, $1.1795 
cation of decimals. 


oo 


820. How many things are to be considered in connection with the subject of 
Percentage? Name them, and define each. What relation subsists between the 
Percentage, Base, and Rate? Show this from Example 1. 


* 


196 PERCENTAGE. 


‘ 


It will be seen from this example that the percentage is the product 
of the base and rate. 


EXAMPLE 2.—What per cent. of $16.85 is $1.1795 ? 
Here the percentage (the product) and the 16.85) 1.1795 (.07 


base (one of its factors) are given, and the rate 11798 
(the other factor) is required. Divide the prod- : 
uct by the given factor, and the quotient will be Ans. 7% 


the required factor (§ 89). 


EXAMPLe 3.—$1.1795 is 7% of what number ? 


Here again the product and one factor are given, 

.O7) $1.1795 and the other factor is required. Divide the product, 

Ans. $16.85 $1.1795, by the given factor, 7%, expressed decimally ; 
and point off the quotient as in division of decimals. 


321. Ruies.—I. Zo find the percentage, multiply the 
base by the rate expressed decimally. 

II. Yo find the rate, divide the percentage by the base ; 
the figures of the quotient to the hundredths’ place inoht- 
sive will denote the rate %, and the remaining figures, if 
any, the decimal of 1%. 

Ill. Zo find the base, divide the percentage by the rate 
expressed decimally. Hence these formulas :— 


PERCENTAGE = Base X RATE 


PERCENTAGE PERCENTAGE 
Rice 2 ee Base tec se ee 


Base Rate 

Proor.—These rules may be used to prove one another. Thus :— 

If the percentage has been found by Rule I, divide it by the rate, ac- 
cording to Rule III., and see whether the given base results. 

If the rate has been found by Rule II., multiply the base by it, accord- 
ing to Rule I., and see whether the given percentage results. 

If the base has been found by Rule III, multiply it by the rate, ac- 
cording to Rule I., and see whether the given percentage results. 


Be very careful to place the decimal point correctly. 


322, EXAMPLES FOR PRAOTIOE. 


1. How much is 154 of £10 4s. 6d.? 


By § 285, £10 4s. 6d. = £10,225. £10,225 x .15 = £1.53376, 
By § 284, £1.538375 = £1 10s. 8d. .4 far. Ans. 


Explain Ex. 2. Explain Ex. 3.—321. Recite the rule for finding the percentage. 
For finding the rate. For finding the base, Express these rules briefly in formulas. 
Show how each operation may be proved. 


a 


EXAMPLES FOR PRACTICE. 197 


2, low much is 50% of £64 18s, 8d. ? Ans. £32 9s, 4d, 
50 % being -4, the shortest way is to take $ at once. 
So, for 3344 take 4. For 1244 take 4. 

For 25%." 4. For 10%; “ +4. 

Mor 20-Fo. &. For8s% 2 ahs. 

For 163% ‘4. HORS ey vs ag: 
3. Find 6% of $1000. Ans. $60.) 12. Find 9% of $995. 
4, 8% of $28.98. Ans. $2,318. | 18. 25% of 78 bu. 2 pk. 
5. £% of £120. Ans. 6s.| 14. 64% of $75. 
6. 42% of 75 gal. Ans, 3.3 gal. | 15. 24% of £10 10s. 
7. 11454 of 8yd. Ans. 11.988in.| 16. 84% of 83 cwt. 8 1b. 
8. 374% of $60.005. Ans. $22.50+.]17. 302% of $122.50. 
9. 20% o0f £10 5d. Ans. £2 1d.| 18. 124% of £8 Is. 4d. 


=" 
S 


. ¥% of 9171 acres. Ans, 30.57 A. 19. 183 % of $240.505. 
. 24% of 50 guin. Ans. 1g. 5s. 3d.| 20. 100% of 16 1b. 5 oz. 1dr. 
21. Find 388% of 4. 97% of 16. 500% of 7. 840% of 284. 
365% of. 92% of 2. Sum of answers, 292.6975. 
22. aan the percentage on $987634.37 at a of the follow- 
ing rates: 4%; 24%; 4%; 84%; $%; 63%; 25%; 412%; 900%; 
43 %. Sum of answers, $18760215.86 +. 
23. A farmer, raising 1097 bu. of wheat, gives 10% of it for 
thrashing, and sells 10% of the remainder. How much is left? 
24, A merchant, who had $6480 invested in business, lost 75 % 
of it. How much did he save? Ans. $1620. 
25, A and B invested $100 each in speculations. <A lost 100% 
of his investment, and B made 200% on his. How much better 
off was B than A on these speculations ? 
26. If 86% of the contents leak out of a hhd. of molasses, how 
many gallons will be left ? Ans. 40.82 gal. 
27. A coal-dealer bought 17180 tons of coal; he sold 624% of 
it at $6.75 a ton, and the rest at $7. How much did the whole 
bring ? Ans. $117597.10. 
28. What is the sum of }% of $40 and .8% of $30? 
29. A California miner, having obtained 15}1b. of gold dust, 
has it melted up and refined. 6% being deducted for the waste 


—_ 
ear 


and cost of refining, what weight should the miner receive ? 
E 


198 PERCENTAGE, 


30. What % is 1 of 20? (Ex. 2, p. 196.) | 40. What % of 4 is 4? 


31. .048 of 240? Ans. .02%.|41. What @% of 3 is 4? 
32. $117 of $900? Ans. 13%.| 42. What @ of 3 is}? 
33. 200 bu. of 50bu.? Ans. 400%. | 43. What% of 1600 is 1000? 
34, 8s. of £100 ?* Ans. .15%.| 44. What @ is $6 of $18? 


85. $2.25 of $112.50? Ans. 2%.| 45. $36.10 of $902.50 ? 
36. 2s. 9d. of £2 15s. ? Ans. 5%.| 46. 76 trees of 400 trees? 
3%..8 3. of 25.1b? Ans. $%.| 47. 1 of 10000? 

38. 1.6 pt. of 40 gal. ? Ans. $%. | 48. 5 mills of 1 dime? 

39. $8.736 of $1248? Ans. .007%.|49. 2 cwt. of 100 tons? 
50. What % of a long ton is a common ton ?* Ans. 892 %. 
51. What % is a pound Troy of a pound avoir.?* Ans. 822%. 
52. What % is an ounce Troy of an ounce avoir.? Ans. 1093 4. 
538. What % is the wine gal. of the beer gal.? Ans. 81434. 
54. A lady divides $300 among her three sons, giving the first 

$100, the second $25, and the third the rest. What per cent. of 

the whole does each receive ? 
55. A person owns a house and lot worth $5500. The lot is 

- worth $1000; what % is that of the value of the house ? 

56. 251s 4% of what? Ans. 625. | 64. $56 is 14% of what? 

57. $10 is 12% of what? $834. | 65. $81.50 is 100% of what? 

58. 9s. is $% of what? £90. | 66. 21c. is .8% of what ? 

59. 17 qt. is 8$% ofewhat? 50gal.| 67. 1200 is 40% of what? 

60. 20c. is .01 Gof what? $2000. | 68. 28.8 bu. is 22% of what? 

61. 1.25]b. is£% of what? 5 cwt. | 69. $456.75 is 105 4 of what? 

62. 1s. is 84% of what? £1 10s.| 70. 1% of 240 is 80% of what? 

63. $40 is 150% of what? $262. | 71. 10% of 8300is5 4% of what? 
72. A farmer keeps 254% of his sheep in one field, 15% in 

another, and the rest, numbering 48, in a third. How many 

sheep has he? Ans. 80 sheep. 
73. A collector, who gets 3 % for his services, makes $33.33 by 
collecting a certain bill. How large is the bill, and how much 
must he pay over to his employer ? Last ans. $1077.67. 
74, Of what number is $% of 90 three hundred 4? Ans. .15. 


* Before dividing, reduce dividend and divisor to the same denomination. 


PERCENTAGE. 199 


75. The deaths in a certain county average 320 a month, and — 
the number of deaths each year is 3% of the population. What 
is the population ? Ans. 128000. 

76. A person worth $40000 gave 30% of it to his son, and this 
amount was 75 % of what his son had before. How much had the 
son after receiving the father’s gift ? Ans. $28000. 


323. To find the base, the rate and the sum or differ- 
ence of the percentage and base being given. 

ExamMpie 1.—A farmer, having a certain number of 
sheep, bought 334% of that number more, and then had 
256. How large was his flock at first ? 


As he increased his fleck by 3344 of orl ae oe 
itself, he must then have had 13344 of 100 + 83§ = 1333% = 


the original number or base. As 256 is 1.334 = 14 = ¢ 
13344 of the base, to find the base, di- 256 + 4 = 192 
vide 256 by 1. 331, or its equivalent > Ans. 192 sheep. 


(Rule III. 321), 

EXAMPLE 2.—A farmer, having a certain number of 
sheep, sold 334% of them, and then had 128 left. How 
large was his flock at first ? 


100 -— 334 = 663 As he sold 334.4% of his flock, he must 

663% = 665 = = 3 have had left 662 ¢ of the original number or 

base. As 128 is "663% of the base, to find the 

128 + % = 192 base, divide 128 by .663 or its equivalent z 
Ans. 199 sheep. (Rule III. § 321). 

Rute.—Divide the given number by 1 increased or 

diminished by the rate expressed decimally, according as 


the sum or difference of the percentage and base ts given. 
77. When I add to a certain number 25 4 of itself, I get 540; 


what is the number ? Ans. 432. 
78. What number is that which diminished by 34 of itself is 
778.69 2 Ans. 782. 


79. A gentleman, having bought a ee spent 10% of the 
purchase price in repairs, and then found that the whole cost was 
$8800? What was the purchase price ? 

828. In stead of the percentage, what may be given, with the rate, to find the 


jase? Explain Examples land 2. Recite the rule for finding the base, the rate 
snd the sum or difference of the percentage and base being given. 


200 PERCENTAGE. 


80. A merchant, having lost 7% of his capital, has $23250 left ; 

what was his capital ? 

81. A farmer set out some apple-trees; 5% of them died the 
next summer, and 3% the following winter; 188 lived. How 
many trees did he set out? Ans. 150 trees. 

82. A lady spent 75% of her money for a cloak, and 5% for 
gloves; she then had $16 left. How much did she have at first ® 


324, Applications OF PrrRcENTAGE.—The rules of 
Percentage are applied in many of the most common 
mercantile transactions. They form the basis of com- 
putations in Profit and Loss, Interest, Discount, Commis- 
sion, Bankruptcy, Insurance, Assessment of Taxes, &c. 


Profit and Loss. 


320. Profit (or gain) and Loss are generally reckoned 
at a certain per cent. of the cost. 

The cost is the base, 

The per cent. of profit or loss is the rate. 

The amouné of profit or loss is the percentage. 


026. Hence, applying the Rules of Percentage (§ 321), 


Prorir or Loss = Cost x Ratr 


Prorir or Loss Prorit or Loss 
APR Wes ee ae Cost = 


Cost Rate 


327. When the cost and selling price are given, their 
difference will be the profit or loss,—profit tf the selling 
price is the greater, loss if the cost is the greater. 

$28. To find the selling price,—avhen there is profit, 
add it to the cost; when there is loss, subtract it from 
the cost. 


824, What is said of the application of the rules of Percentage? In what doa 
they form the basis of computations ?—325. Tow are Profit and Loss generally reck- 
oned? What corresponds to the base? What, to the rate? What, to the percent- 
age?—326. Give the formulas that apply.—82T. When the eost and selling price are 
given, what will their difference be ?—328. How do you find the selling price, whee 
shere is profit? When there is loss? 


PROFIT AND LOSS. 201 


EXAMPLES FOR PRAOTICE. 


Vind the pRoFIT or Loss, 
1. On goods that cost $145, sold at 8% advance. Ans. $4.35 pr. 
2. On goods costing £2500, sold at 444 loss. Ans. £112 10s. 
3. On furniture bought for $850.75, sold at 7% below cost. 
4, On paper costing $1485.50, and sold at a profit of 15 %. 
5. On coal bought for $9020, and sold at a loss of 6} 4. 

. On tea sold at $% below cost, which was $666.66. 

Find the serine price of goods, 3 
7. Bought at $88.65, sold at 834% below cost. Ans. $85.695. 
8. Bought at £120, and sold at 8% advance. Ans. £129 12s. 
9. Sold at 20% below cost, bought for $18000. 
10. Sold at 102% above their cost, which was $5050. 

Find the rate % of:profit or loss on goods, 
11. Bought for $13000, sold at a profit of $292.50. Ans. 24%. 
12. Bought for ¢80, sold for $60. 3 Ans. 25%. 
183, Bought for $113.25, sold so as to gain $113.25. 
14, Bought for $5601.30, sold so as to lose $2800.65. 
15. Bought for £250, sold for £200 (§327). Ans. 204 loss. 
16. Bought for $1250, sold for $1375. Ans. 10% prof. 
17. Sold for $1090, bought for $1000. Ans. 9% prof. 
18. Sold for $245.18, bought for $235.75. 
19. Bought for $800, and sold for $894.40. 
20. Bought for $740, and sold for $627.15. 
21. Bought for $815, and sold for $220.05. 
22. Bought for $350.50, and sold for $701. 
23. Sold for $540, at a profit of $40. Ans. 8%. 
24. Sold for $600.354, at a loss of $26.64%. Ans, 444%. 
25. Sold for $200, at a loss of $100. 

Find the cost of goods, 
26. Sold at a profit of $40, being 204 on the cost. Ans. $200. 
27. Sold at 7% below cost, at a loss of $350. Ans. $5000. 
28. Sold at 124% above cost, at a profit of $240. Ans. $1920. 
£9. Sold at a loss of $53, being 4 % of the cost. 
30. Sold at 4% above cost, at a profit of $10.50. 


=r) 


202 PROFIT AND LOSS. 


329. Zo find the cost, when the selling price and rate 
of profit or loss are given. 


KixampLe 1.—A sold a horse for $175, and by so doing 
gained 40%. What did the horse cost ? 


This question is analogous to Example 1, § 323, under Percentage. 
As he gained 40% of the cost, the selling price all 
must have been 100 + 40, or 140,% of the IOP , pit ae 
cost. The question then becomes, $175 is 175 + 1.40 = 125 
140% of what number? (Rule IL, $321.) Ans. $125. 

Exampir 2.—A sold a horse for $175, and by so doing 
lost 40%. What did the horse cost ? 

100 — 40 = 604% This is analogous to Ex. 2, $323. As he 
175 + .60 = 2912 lost 40 % of the cost, the selling price must have 


: x been 100 — 40, or 60, % of the cost. The ques- 
Ans. $291.066% tion then becomes, $175 is 60 % of what number? 


Rurze.— Divide the selling price by 1 increased by the 
rate of profit, or diminished by the rate of loss, expressed 
decimally. 


81. By selling a house and lot for $5790, the owner lost 34 4. 


What was their cost ? Ans. $6000. 
32. Sold 517 barrels of flour for $8, 10 a barrel, at a profit of 
8%. What was the whole cost? Ans. $3877.50. 


33. Sold 1100 tons of coal for £1861 5s., thereby losing 1%. _ 


What was the cost per ton ? Ans. £1 5s. 
34, Some linen was sold for 614c. a yd., at a loss of 54. What 
was the cost.of 7 pieces of this linen, averaging 13 yd. to the 
piece ? Ans. $59.15. 
35, Sold a book-case for £15, and some books for £33 Qs. 6d., 
and thereby gained 204%. What was the cost of case and 
books ? Ans. £40 4d. 3 far. + 
36. D bought 5000 bu. of corn, but lost 10% of it by fire; he 
sold what was left for $3408.75, and by so doing gained 14% on its 
cost. What did he give for the 5000 bu. ? Ans. $3750. 
87. Selling price, $4778.75; gain, 4%; required, the cost. 


329. Explain Examples 1 and 2. Recite the-rule for finding the cost, when the 
sclling price and the rate of profit or loss are given. 


PROFIT AND LOSS. 72038 


330, Zo jind the rate ef profit or loss at a proposed 
selling price, when the actual selling price and rate of 
profit or loss are given. 

Ex.—If, by selling a cow for $60, I gain 20%, what @ 
would I have gained or lost by selling her for $25 ? 


First find the cost, § 329: $60 + 1.20 = $50, cost. 
Then find the gain or loss at the 

proposed selling price : $50 — $25 = $25, loss. 
Find the rate, by dividing the loss 

by the cost: 25 +50 = 504 loss. Ans. 


Rutz.—from the selling price obtain the cest (§ 329) 5 ; 
then find the gain or loss at the proposed selling Price by 
subtraction, and divide it by the cost. 


38. A profit of 4% is realized by selling some cloths for $228.80 ; 
had they been sold for $215.60, what 4 would have been gained or 
lost ? Ans. 2% lost. 

59. Some grain is sold for $1335, at a loss of 11%; what amount 
would have been gained or lost, and what @, if it had been sold 
for $3000 ? Last ans. 100% gd. 

40. By selling some goods at $1587.90, a profit of 123% was 
realized ; what per cent. would have been gained or lost, if they 
had sold for $1651.65 ? . Ans. 21% gd. 

41, 24% was lost by selling a farm for $13650; what % would 
have been gained or lost by selling it for $13986 ? 

42. If by selling some wood for $850 I made 100%, what % 
would I have gained by selling it for $1275 ? 

43. Sold a house for $1000, thereby making $200 ; what would 
I have had to sell it for, to gain 50% ? 

44, By selling some goods for $4759.79, 4+ of 1% was gained. 
What would these goods have had to be sold for, to realize a profit 
of 7%? Ans. $5085.71. 

45. A merchant bought 320 barrels of flour at $7.50 a barrel, 
and sold them at a loss of 10%. How much did he lose? 

320. Explain the given example. Recite the rule for finding the rate of profit 


er loss at a proposed selling price, when the actual selling price and rate of profit er 
oss are given. 


204 PROFIT AND LOSS. 


46. Bought 800 yd. merino at $2.25 a yd., and sold the same 
at $2.50 ayd. How much was gained, and what % ? 

AY. Twenty-five cords of wood were bought at $4.50 a cord, 
and sold at an advance of 25%. 404% of the bill was paid in cash; 
how mutch remained to be paid? Ans. $84.375. 

48, Bought a lot for $600, fenced if for $50, and built a house 
on it for $1550. Sold the whole at a profit of 84%; what did it 
bring ? Ans. $2381.50. 

49, A buys $1900 worth of goods, which he sells to B at a 
gain of 5%. B sells them to C at a profit of 5%, and © sells them 
to D at a like profit. What did they cost D? Ans. $1157.625. 

50. § sells T some goods that cost him $1480, at a loss of 
34%. A few days afterwards, T sells them back again to § at a 
gain of 34%. How much less does S pay for them the second 
time than the first ? Ans, $1.81 +. 

51. P buys an article for £50 18s. 6d., and sells it to Q at a 
profit of 10%. Q in turn sells it to R at a loss of 10%. What % 
of the original cost does R pay ? Ans, 99%. 

52. If a person buys 600 barrels of flour at $9.25 a barrel, and 
sells 834% of the same at a profit of 10%, and the rest at a profit 
of 124%, now much will he receive in all, and what 4 will he gaim 
on the whole? Last ans. 113%. 

58. Bought 8000 bu. of wheat at $1.60 a bushel. Sold 10 per 
cent. of it at a loss of 84%, 50 per cent. of it at a gain of 10%, and 
the rest at a gain of 2%. How much was made on the whole, and 
what per cent. ? Ans. $264, and 54%. 

54, Sold some muslin for $199, 50, at a loss of }%; some linen 
for $148.50, at a loss of 1%; some cloth for $520, at a profit of 
4%, What did the muslin, linen, and cloth cost ? Ans. $850, 

55. Sold a horse for $198, at a loss of 10%. Bought 8 cows for 
$135. What must I sell the cows for apiece, to make up the loss 
on the horse and $44 besides ? Ans. $67. 

56. The difference between 50% and 71% of a certain number 
is 525. What is the number? 

57. A house that cost $5000, was repaired at an expense of 
$1000. It was then sold for $7500; what was the gain or loss 73 


INTEREST. 205 


CHAPTER XX. 
INTEREST, 


331, Interest is what is paid for the use of money. 

The Principal is the money used, for which interest i¢ 
paid. The Rate is the number of hundredths of the prin- 
cipal paid for the use of the principal for a certain time, 
usually for a year (per annum). It is written and oper- 
ated with as so many per cent. When no time is men- — 
tioned with the rate, a year is meant. : 

The Amount is the sum of the principal and interest. 


I borrow $100 for a year, and pay $6 for its use; the Principal is 
$100, the Interest $6, the Rate 64, the Amount $106. 


332, Interest is dieanprimted as Simple and Compound. 
It is called Simple, when reckoned on the principal only ; 
Compound, when allowed on interest as well as principal. 
When the word ¢nterest is used alone, Simple Interest is 
meant. 


333. There is a rate of interest fixed by law, mlied the 
Legal Rate, for cases in which no other rate is specified. 
Parties may always agree on a lower rate than the Legal 
Rate, and in some of the states on a higher one; but 
there is generally a limit-fixed, beyond which the taking 
of interest is forbidden under certain Lorlaey ne of: 
fence being called Usury. 

The legal rate in England and France is 5%; in Can- 
ada, Nova Scotia, and Ireland, 6%. In all of the United 
States it is 6%, except the following: Louisiana, 5%; New 
York, Michigan, Wisconsin, Minnesota, South Carolina, 

831. What is Interest? Whatis the Principal? What is the Rate? How is 
the Rate written and operated with? What is the Amount ?-- Illustrate these defi- 
nitions. —832. Ilow is interest distinguished? When is it called Simple? When, 
Compound ?—333. What is meant by the Legal Rate? May the parties agree on # 
lower rate than the legal one? Ona higher one? Whatis Usury? In what coun- 


tries is the legal rate 5 per cent.? In what, 6 per cent.? In which of the United 
States is it 5 per cent.? In-which,7 per-cent.? In which, 8 per cent.? In whieh, 


206 INTEREST. 


and Georgia, 7%; Alabama, Florida, Mississippi, and 
Texas, 8%; California and Kansas, 10%; Oregon, 124 4%. 

334, Interest is an application of Percentage, the ad- 
ditional element of é¢me being introduced. The principal 
is the base ; the interest is the percentage, reckoned at a 
certain rate, for a certain time. 


To find the Enterest. 


335, Cass 1L—To find the interest for any number of 
years, when the principal and rate are given. 
Ex. 1.—What is the interest of $124.50, for 1 year, 


at 6%? 

That is, what is 6 %, or +83, of $124. 50? Taking $124.50 Prin. 
Tee is equivalent to BAB by zé5. Hence, 6 Rate. 
multiply the principal by .06. $7.4700 Int. 


Ex. 2.—Find the amount of $124.50, at 64, for’S years. 
Find the interest for lyr. as above, 


re : 
ee ae $47.47. For 5 yr. it will be 5 times $7.47 ; 
i sa aia and, as the amount is required, add the 
7.4700 Int. 1 yr. principal to the last product. 
5 In stead of multiplying by the rate and 
37.3500 Int. 5 yr. years separately, it sometimes saves work 


“ to multiply by their product. Thus, in 
sey Agee : Example 2, it would be shorter to multi- 
$161.85 Amt. 5 yr. ply by .80 than by .06 and 5. 


Ruie.—Multiply the principal by the rate per annum 
expressed decimally, and that product by the number of 
years. 


Aliquot parts of a year may be expressed Saran Thus, 5 yr. 
6mo. = 5zyr. See Table, page 186. 


336. So, when the rate is given by the month, the interest may be 
found for any number of months, by multiplying the principal by the rate 
per month expressed decimally, and that product by the number of months. 


30%. Hor the Amount, add the principal to the interest. 


10 percent.? In which,124percent.? In the rest ?—334. Of what is Interest an ap- 
plication? What additional element is introduced ?—835. What is Case I.? Explain 
Example 1. Ge through Example 2. Recite the rule. How may aliquot parts of 
a year be expressed ?—336. When the rate is given by the month, how may the in- 
terest be found for any number of menths ?— 337. How is the amount found? 


orn . =) 


Lod 


INTEREST, 207 


EXAMPLES FOR PRAOTIOR. 


. Find the interest of $1, at 844, for 3 years. Ans. 10e. 
. Find the amount of $540, at 7%, for 9yr. Ans. $880.20. 
. Kind the interest of $90, at 44.4, for 6 yr. Ans. 24.30. 
. Find the amount of £1400, at 8%, for 24 yr. Ans. £1680. 
Find the interest of $825, at 64.4%, for 4 yr. Ans. $206.25. 
. What is the interest of $33120.01, for 5 yr., at 6%? 

. What is the interest of $987.41, for 13 yr., at 7%? 

. Find the interest of $69582.57, at 5%, for 24 years. 

. Find the amount of $9812.17, at 424, for 4 years. 

10, Find the amount of $700, at 64, for 2yr. 6 mo. (24 yr.). 
11. Find the amount of $820, at 34, for 4yr. 4mo. (44 yr.). 
12. Find the amount of $660, at 5%, for 3 yr. 38 mo. 

13. What is the interest of $60.50, for 8 months, at 1% a 


OO TO OP oo to 


month? (See § 336.) Ans. $1.815. 
14. What is the amount of $12198.75, for 2 months, at 3% a 
month ? Ans. $12881.78. 
15. What is the interest of £600, at $% a month, from Jan. 1 
to April 1 of the same year ? Ta hee eek 
16. What is the amount of $8250, from April 8, 1861, to April 
8, 1866, at 52% per annum ? Ans. $10621.875. 


17. Borrowed, Jan. 1, 1865, in California, $900 (no rate speci- 
fied). What amount must be repaid, Jan. 1, 1866? Ans. $990. 
18. A owes B interest on $450, from Feb. 2 to Oct. 2; B owes 
A interest on $575, from April 2 to Oct. 2. What is the balance 
of interest, and to whom is it due, the rate being 4% a month? 
Ans, [5%., to B. 
19. What is the interest on $68.40, at 45%, for 4yr. 2mo. 
(41 yr.)? On $4712, for 6yr. 3mo., at5%? On $2688.88, at 63%, 
for lyr. 6mo.? On $1268.25, for 5mo., at 1% a month? 
Sum of answers, $560.1783. 
20. Loaned, New York, Feb. 1, 1864, $1050. What amount 
should I receive for loan and interest, March 1, 1866? 
21. CO, living in Canada, owes $500 with interest for 3 yr. 2 mo. 
He pays $550 on account; how much remains due ? 


208 INTEREST. 


338, Casn IL—7o find the interest, at 6 per cent., for 
years, months, and days. 

1. For a given time and rate, the interest or amount 
of any principal is as many times greater than the inter- 
est or amount of $1, as the principal is greater than $1. 


Thus, the interest of $50, for 5 mo., at 6%, is 50 times the interest of 
$1, for 5mo., at 6%. The amount of $60, at 7%, for 30 days, is 60 times 
the amount of $1, at 74%, for 30 days. 

2. The interest of $1, at 6%, is 6 cents for 1 year. 
Hence it is 1 cent for every two months, and 1 mull for 
(45 of 2 months, or) 6 days. 


6 days are +1; of 2 months, if 80 days are allowed to the month, ac- 
cording to general usage in the United States. Each. day’s interest is 
thus made 3}, in stead of 3}5, of 1 year’s interest; it thus exceeds the 
exact interest by 325, or 7/5, of itself. To find the exact interest for any 
number of days, see § 342. 


ExamMPLe 1.—What is the interest of $1200, at 6 %, for 
3 yr. 7mo, 18 da. ? 
First Method.—First find the interest of $1, at 6 %, for the given time. 


~ As the interest is 1 cent for every 2 months, 1b 
for 3 years 7 months, or 43 months, it will be 4 of 215 
43 cents, or $.215. As the interest is 1 mill for 6 -003 
days, for 18 days it will be 4 of 18 mills, or $ .003. $ .218 
Adding §.215 and $.003, we find the interest of 1200 


$1 for the given time to be $.218. For $1200 it eA 
will be 1200 times as much, or $261.60. $261.600 Ans. 


Second Method.—First. find the interes 
$1200 of $1200 for 1 yr., then for 8 yr., as in Case 
.06 I. For the months and days apply the prin- 
6 mo. = 4] 72.00 ciples of Practice, § 307. 

3 7 mo. are not an aliquot part of 1 yr., 
but 6 mo. = 4yr.; therefore, for 6 mo. take 
216.00 4 of 1 year’s interest, and for 1 mo., which 

36.00 remains, take 4 of the interest for 6mo. 18 


1 mo. 


= 3 6.00 ‘days are not an aliquot part of 1mo., but 
15 da. = 4 3.00  15da. = 4mo.; therefore, for 15 days take 
3 da. = +4 | .60 4 of 1 month’s interest, and for 8 days, which 


————— 


Ans. $261.60 remain, take + of the interest for 15 days. 


Finally, add the several items of interest. 


338. What is Case II.? To what is the interest or amount of any principal for 
a given time and rate equal? Give examples. For how long a time will the inter- 
est of $1, at 6 per cent., be 1 cent? 1 mill? How does the interest for 1 day thus 


INTEREST AT SIX PER CENT. 209 


339. Ruty—l. Zo 4 the number of months written 
as hundredths add 4 the number of days written as thou- 
sandths, and multiply the sum and the given principal 
together ; the product will be the interest. Lf the amount 
is required, add 1 to the above sum before multiplying. 

2. Or, find the interest first for the given number of 
years, as in § 335, then for the months and days by taking 
ihe necessary parts, and add the results. 


The first method is generally shorter and easier. 


EXAMPLE 2.—W hat is the amount of ere 
$66.60, at 6%, for 1 year 11 mo. 11 da. ? ae 
1 yr. 11 mo. = 28 mo. Writing 4 the num- 39960 
ber of months as hundredths, we have .115. 6660 
Writing 4 the number of days as thousandths, 6660 
we have .0018. As the amount is required, we 6660 
add in 1. .115 + .0018 + 1 = 1.116%. Mul- yee 
tiply the principal by 1.1168. Ans. $74.388110 


EXAMPLES FOR PRAOTIOE. 


At 6 per cent., required the 

. Interest of $49.37, for lyr. 1mo. 15 da. Ans. $3.83 +. 
. Amount of $341.18, for Tyr. 9 da. Ans. $484.916 +. 
. Amount of $591.03, for 4 yr. 8 mo. 7da. Ans. $742.48 +. 
. Interest of $0.134, for 4months 3days. Ans. $.0027+4+. 
. Amount of $7.50, for 7 months. Ans. $7.76 +. 
. Interest of $871.01, for 4 years 1l5days. Ans. $89.969 +. 
. Interest of $57.92, for 3 yr. 7 mo. 9 da. Ans. $12.53968. 
. Amount of $329, for 5 years 18 days. Ans. $428.41 +. 
. Amount of $47.39, for 1 year 7 months. 

10. Interest of $2250, for 2 yr. 2mo. 24 da. 

11. Interest of $5762, for 6 yr. 4mo. 19 da. 

12. Amount of $840.75, for 11 months 21 days. 

18. Interest of 98.76, for 3 yr. 5mo. 22da. Ans. $20.60792. 


Oo OT dD iP oo NS we 


computed compare with the trne interest? Go through Example 1 according to 
each method.—839. Recite the Rule. Which method is preferred? When the 
rmount is required, what must be done? Explain Example 2. 


210 INTEREST. 


14. Interest on $718, from April 19 to Aug.3 following. From 
Oct. 29, 1865, to Feb. 11, 1866. Sum of answers, $24.65 4+. 
15. Interest of £500, for 2 yr. 4mo., 12 da. Apert. 


Compute the interest on pounds as on dollars. A decimal in the answer must 
be reduced to shillings, &e. 


16. Amount of £2500, for 1 year 9 months 18 days. 

17. Interest of £480, for 1 yr. 3mo. 20 da. Ans, £37 12s. 

18. Amount of £60, for Syr. 6mo. 2da. Ans. £90 12s. 4d.+. 

19. P owes Q $975, with interest for 1 yr. 10 mo. 10da.; Q 
owes P $720, with interest for 2yr. 25da. The rate being 67%, 
what is the balance, and to whom is it due? Ans. $2'74.475, to Q. 

20. A merchant collects the interest on $400, at 7%, for lyr. 
§mo.; on $220, at 6%, for 8mo, 8da.; on $694.10, for 2yr. 2 da., 
at 6%; and on $1180.50, for 26 days, at 6%. How much does he 
‘collect in all ? Ans. $1389.782 +. 


340. Merchants often have to cast interest, at 64%, for 
30, 60, and 90, also for 33, 63, and 93 days. The follow- 
ing short methods can be used mentally :— 

For 60 days, simply move the decimal point in the 
principal two places to the left,—for this will be multi- 
plying it by .01, the interest of $1 for 60 days being $.01. 

For 30 days, take 4 of this result. 

For 3 days, take #, of the interest for 30 days,—that 
is, move the decimal point one place to the left. 

Combine these results as may be required. 


Examprie.—Required the interest of $560, at 6%, for 
80, 60, 90, 83, 63, and 93 days. 


Int. 60 days, $5.60 ( Int. 90 days, $5.60 + eee = $8.40 
Int. 30d 9.80 Then Int. 33 days, $2.80 + $0.28 = $3.08 
boa D4 Int. 63 days, $5.60 + $0.28 = $5.88 

Int. 3 days, $ .28 Int. 93 days, $8.40 + $0.28 = $8.68 
At 6 per cent., what is the interest of 

21. $700 for 60 days? For 83 days? For 90 days? 

22. $1200 for 30 days? For 60days? For 90 days? 

23. £1000 for 63 days? For 90 days? For 93 days? 


INTEREST FOR ANY TIME AND RATE, 211 


24, $74.75 for 60 days? For 63days? For 93 days? 
25. $180.90 for 83 days? For 63 days? For 93 days? 
26. $2000.50 for 80 days? For 90 days? For 33 days? 


341. Case IIL—Zo find the interest, at any rate, for 
years, months, and days. 


Ex,—Find the interest of $126, at 7%, for 2 yr. 5mo. 6 da. 


4 number of months, written as hundredths, 145 
4 number of days, written as thousandths, 001 
$120 x .146 = $17.52 146 
Interest of $120, for the given time, at 6 4, $17.52 
For 7 4, add to the interest at 6 @, 4 of itself, 2.92 
(for 7 = 6 + 4 of 6). $20.44 Ans. 

4 Or, we may find the in- 

oe Rue terest at once at 7 %, for 1 

ae = ; yr.; then for 2 yr., by mul- 

4mo. = $| $8.40 Int. 1 yr. tiplying by 2; then for the 

| 2 months and days, by taking 

$16.80 Int. 2 yr. parts. For 4 mo., take 4 

9.80 Int. 4 mo. of 1 year’s interest; for 1 

1lmo. «= + 70 Int. 1 mo mo., take + of 4 months’ 

6 da. = 4 "44 The 6 da : interest; for 6 days, take 4 


of 1 month’s interest. For 
Ans. $20.44 Int. 2y. 5m. 6d. the whole, add these parts. 


Ruuz.—l. Find the interest at 6%, and add thereto, or 
subtract therefrom, such a part of itself as must be added 
to or subtracted from 6 to produce the given rate. 

For 7%, add 4(7 = 6++0f 6). | For 52, subt.3 (5 = 6 — tof 6). 
For 8%, add (8 = 6 + of 6). | For 414, subt. £(44 = 6 — $ of 6). 
For 9%, add 4(9 = 6+40f6). | For 42, subt.4(4 = 6 — 4 of 6). 
For 104, add 4 (10 = 6 + 2 of 6). | For 3 4, take 4 the interest at 6 ¢. 


2. Or, find the interest at the given rate, for the given 
number of years, as in § 335; then for the months and 
days, by taking the necessary parts ; and add the results. 


841. What is Case IIL? Give both solutions of the Example. Iecite the rule. 
Yor 7 per cent., what must we do, and why? For4percent.? For 10 per cent.? 
For 3 per cent.? For 9 per cent.? For 43 percent.? For8percent.? For 5 
per oeng, 7 


212 | INTEREST. 


EXAMPLES FOR PRAOTIOE. 


What is the interest (by either or both of the methods given 
in the preceding Rule) of 


1. $5.87, for 4 years 12 days, at 8%? Ans. $1.73 +- 
2. $40.17, for 8 months 18 days, at 3%? Ans. 36c. + 
3. $37.13, for 5 months 12 days, at 44%? Ans. T5e. + 
4, $194.10, for 1 yr. 7 mo. 13 da., at 7%? Ans. $22. + 
5. $821.21, for 5 yr. 9 mo. 21 da., at 9%? Ans. $167.91 +. 
6. $9872.86, for lyr. 5mo. 11da., at 7%? Ans. $1000.175 +. 
7. $999.99, for 11 months 29 days, at 5%? Ans. $49.86 +. 
8. $27541.03, for yr. 10mo. 22da., at 7%? Ans. $5580.11 +. 
9. $187.50, for 6 mo. 10da., at 64%? (Add 5.) Ans. $4.717 +. 


= 
=) 


. $4650, for 3 yr. 4 mo. 12 da., at 7%? Ans. $1095.85. 
. $2000, for 83 days, at 10%? For 63 days? 

. $11500, for 60 days, at 4%? For 90 days? 

. $8260, for 3 yr. 29 da., at 54%? (Subtract +4.) 

. $428.07, for 1 yr. 1 mo. 1 da., at 7%? 

. $.75, for 10 yr. 10 mo. 10 da., at 5%? 

16. A, living in New York, owes B $625, with interest from 
Jan. 1 to Sept. 15, no rate specified. He pays on account $540.25, 
how much remains due? ~ Ans. $115.618. 

17. What is the amount of $469.10, for 3 yr. 2mo., at 7%? 
For lyr. 20 days, at 4%? For 11 mo. 19da., at5%? For 6 mo. 
6 da., at 8%? Sum of answers, $2042.318 +. 

18. A merchant living in Mississippi collects $100, with inter- 
est for 3 yr. (no rate specified); $427.50, with interest for 8 mo. 
9 da., at 7%; $1100, with interest for lyr. 18da., at 6%. How 
much does he collect in all? Ans. $1741.50. 


ay eee | 
oF wt DY Ke 


342, Caszr 1V.—To find the exact interest for days. 

The exact number of days between two dates within a 
year of each other can be found by the Table on page 156. 

Each day being 1, of 1 year, the exact interest for 


342, What is Case [V.? How may the exact number of days between two dates 
within a year of cach other be found? What fraction of 1 year’s interest will the 
ext interest for any number of days be? Recite the rule. Solve the Example. 


EXACT INTEREST FOR DAYS. 213 


any number of days will be as many 365ths of 1 year’s 
interest as there are days. Hence the following 


Rourtz.— Multiply the interest for 1 year at the given % 
by the number of days, and divide the product by 365. 

Examprie.— W hat is the exact interest of $37.37, from 
May 3, 1865, to Dec. 27, 1865, at 7%? 


1 year’s interest = $37.37 x .07 = $2.6159. 
By Table, p. 156, we find the number of days to be 288. 
We must therefore take 22? of $2.6159. 
$2.6159 x 238 = $622,5842, $622,.5842 — 865 = $1.705. Ans. 


1. What is the exact interest of $100, at 6%, from Jan. 18 to 


Noy. 15, it being leap year ? Ans. $5.047. 
2. What is the exact interest of £1000, from June 20 to Aug. 
18, at 7%? Ans. £10.856 = £10 7s. 1d. 1 far. + 
3. What is the exact interest of $730, from July, 4 to Dec. 25, 
at 6%? Ans. $20.88. 
4, What is the exact interest of $2160, from March 10 to Dec. 
1, at 5%? What is the amount? Ans. Amt., $2238.71. 


5. What is the exact interest of $21450, at 8%, for 20 days? 
6. What is the exact interest of £4500, at 44%, for 25 days? 


343, Cas—E V.—To find the interest or amount of 
pounds, shillings, pence, and farthings, at any rate, for 
any time. 


£84,525 

Examprte.—What is the inter- .04 
est of £84 10s, 6d., at 4%, for 1 yr. 8 = } | 3.88100 
3 mo. ? 84525 
For convenience of multiplying and divid- me 
ing, we reduce 10s. 6d. to the decimal of a AEE CES 
pound, § 285. The principal thus becomes s. 4.52500 
£84,525. Now, proceeding as in Federal 12 
Money, we find the interest to be £4.22625,— d. 6.30000 
or, reducing the decimal to shillings, &c., § 284, 4 


£4 4s. 6d. 1.2 far. Py a) Eee 
far. 1.20000 


343. What is Case V.? Go through and explain the given Example. Recite the 
rule for finding the interest or amount of pounds, shillings, &¢., at any rate, for 
any time, 


214 INTEREST, 


Ruity.— Reduce the shillings, &c., of the principal to 
the decimal of a pound, find the interest or amount as in 
Federal Money, and reduce the decimal in the result to 
lower denominations. — 


344, If the rate is 5¢ and the time 1 year, the interest is readily 
found by taking 1s. for every pound of the given principal, 3d. for every 
bs., and 1 far. for every 5d. For, 5% is #o; and Is. is #5 of £1, 3d. is 
zy of 5s., 1 far. is zy of bd. 

Exampie.—Required the interest of £86 17s. 6d., at 54, for 1 year. 

1s. on £1 of the given principal gives 86s, = £4 6s. 

3d. for every 5s. 17s. 6d. + 5s. = 384 84x 3 = 103d. 

| Ans. £4 68. 1034. 


EXAMPLES FOR PRACTICE. 


Find the interest of 

. £760 5s, 6d., at 5%, for Qbyr. Ans. £88 13s, 11d. 2.8 far. 
. £1 7s. 6d., at 44%, for 2yr. 6 mo. Ans. 3s. 1d. 4 far. 
. £8260 18s., at 34%, for 21 da. (§ 842). Ans. £16 12s, 8d. + 
£275 10d., at 4%, for 5yr. 10 mo. Ans. £64 3s. 6d. + 
. Find the amount of £7 15s., for lyr., at 5% ($3844). 

. Find the amount of £42 2s. 6d., for 1 yr., at 5%. 

. Find the amount of £88 7s. 6d., for 1 yr., at 52%. 

. Find the amount of £68 12s. 6d., for lyr., at 5%. 

9. Find the amount of £100 15s. 8d., at 43%, for 2yr. Tmo. 


DaNarar wh 


345, Observe that whenever the product of the rate 
per annum and the number of years is 100 (or 1, if the 
rate is expressed decimally), the interest equals the prin- 
cipal. The interest of $75, at 5%, for 20 years, is $75 ; 
since 5 x 20 = 100 (or .05 x 20 = 1). 

346. In stead of computing interest by any of the 
- methods that have been given, many use Interest Tables. 
These being constructed for different principals, rates, and 
periods of time, the required interest is found in some cases 
by a simple reference, in others by an addition of items. 


344. If the rate is 5 per cent. and the time 1 year, what short method may be 
used? Explain the principle on which this method is based.—845, Under what cir- 
cumstances does the principal equal the interest? Give an example.—346, In stcad 
of computing interest, what do many use? 


INTEREST. 218 


347. To find the Rate, 
the principal, interest or amount, and time, being given. 


Ex. 1.—At what rate will $400 yield $55 interest, in 
2yr. 6 mo. ? 


rn ur: . The interest of $400, at one per 
eel A ee cent. for 2yr. 6 mo., is $10. To 


$4 x 24 = $10, int. 24 yr. produce $55 interest, the rate must 


$10) $55 be as many times 1% as $10 is con- 
tained times in $55, or 54. Ans. 
54 Ans. 5h d. ’ 


Ruie.— Divide the given interest by the interest of the 


principal, for the given time, at 1%. 
348, If, in stead of the interest, the amount is given, subtract the 
principal from it, to find the interest, and proceed as above. 


349, Prove by trying whether, at the rate found, the principal vill 
produce the given interest in the given time. 


Ex. 2.—At what rate will $630 amount to se0e 28, in 
9mo., 18 da. ? 


Find the interest of $630, for 9 mo. 18 da., at 1 7. 
First find it at 6 4%, accord- i % number of months, .045 
ing to $339: & number of days, .003 
At 14 it will be ¢ as much as at 64: 6) .048 
Interest of $1 for the given time, at 1 Z, $ 008 
For $630 it will be 630 times as much: $.008 x 630 = $5.04 
The interest is $665.28 — $630 = $35.28. 
$35.28 + $5.04 = 7. Ans. 7 4. 
Proor.—Will $630, at 7%, amount to $665.28 in 9 mo. 18 da. ? 


EXAMPLES FOR PRACTIOE. 


Prove each example, § 349. At what rate will 


J. $530, in 8 yr. 6 mo., yield $92.75 interest ? Ans. 5%. 
2. $4070 yield $91.575 interest quarterly ? Ans. 9%. 
3. $100, in 9 mo. 10 da., yield $3.50 interest ? Ans. 44%. 
4, £6000, in 5 yr. 20da., amount to £7820? Ans. 6%. 
5. £2600 yield £104 friterést semi-annually ? 


6. At what rate will $1250, in 60 days, amount to $1264. 58h? 


347, To find the rate, what must be given? Analyze Example 1. Recite the 
rule for finding the rate.—348. If the amount is given, in stead of the interest, what 
must you do ?—349. How may this operation be proved? Go through Example 2 


216 INTEREST. 


350, To find the Time, 
the principal, interest or amount, and rate, being given. 


Ex.—In what time will $400 yield $55 interest, at 53%? 


The irterest on $400 for “he ; 
one year, at 544, is $22. To $400 x .055 = $22, int. 1 yr. 
produce $55 interest, will re- $55 -- $22 = 24. Ans. 24 yr. 
quire aS many years as $22 is Proor. : SORE 
contained times in $55, or 24. BOOK aie fake * Baha s 

Rvie.— Divide the given interest by the interest of the 
principal, at the given rate, for 1 year. 

. A decimal in the quotient must be reduced to months and days, § 284. 


Prove by trying whether, in the time found, the principal will produce 
the given interest at the given rate. 


EXAMPLES FOR PRACTICE. 


Prove each example. In what time will 


1. $4070, at 9%, yield $91.575 interest ? Ans. 3 mo. 
2. $530, at 5%, yield $92.75 interest ? Ans. 3 yr. 6 mo. 
83. $100, at 444, yield $3.50 interest? Ans. Z yr. (9mo. 10 da.) 
4, £6000, at 6%, amount to £7820? Ans. 5S yr. 20d. 
Find the interest ;: £7820 — £6000. Then proceed as above. 

5. $820, at 53%, amount to $857.58% ? Ans. 10 mo. 


6. $1250, at 7%, amount to $1264.58% 2 

7. $700, at 7%, amount to $785.75 ? 

8. £680, at 8%, amount to £950? 

9. How long will it take $230, at 4%, to yield $230 interest,— 
that is, to double itself? 


351. In Example 9, the interest equals the principal. 
Hence, the product of the rate per annum and number of 
years must be 100 (§ 345); and, to find the years, we may 
at once divide 100 by the rate. 100+ 4 = 25yr. Ans. 

Were the years given, and the rate required, we should 
divide 100 by the number of years. 100+ 25=44%. Ans. 
850. To find the time, what must be given? Analyze the example. Recite the 


rule for finding the time. Tlow may the operation be proved? What must be done 
with a decimal in the quotient ? 


INTEREST. She 


10. How long will it take $600 to amount to $1200, at 6%? 

11. How long will it take $43.50 to double itself, at 7%? <At 
8%? At8%? At4t¢? AtdG? At9G? 

12. In what time will $150 double itself, at 1%amonth? <At 
fof 1%amonth? At 2% a month? 

13. At what rate will $60 double itself, in 5yr.?2 Ans. 20%. 

14. At what rate will $75 amount to $150, in 12 yr. 6 mo. ? 

15. At what rate will $12.125 amount to $24.25, in 20yr.? In 
16yr.? In i2yr.? In 10 yr.? 


352. To find the principal, 
the interest or amount, time, and rate, being given. 


Ex. 1.—What principal will, in 2 yr. 6 mo., at 534, 
yield $55 interest ? 


The interest on $1, for 1 yr., we : 
at 54%, is $.055; for 2 yr. 6mo., $1 x .055 = $.055 
it is 24 times $.055, or $.1375. $ .055 x 2.5 = § .1375 
To produce $55 interest, the prin- $55 =- $ 13875 = 400 
cipal must be as many times $1 Ans. $400 
as $.1375 is contained times in Re 
$55, or 400. Ans. $400. PRooF. $400 x .055 x 2.5 a $55 


Ex. 2.—What principal will, in 9 months 18 days, 
amount to $665.28, at 7%? 


4 of 9, as hundredths, .045 ¢ 
4 of 18, as thousandths, 003 The amount of $1, for 9 mo. 18 da., 


——_ at 74, is $1.056. To make the amount. 
ne a MA i $665.28, the principal must be as many 
6 Of $.048, 3 .008 times $1 as $1.056 is contained times in 


donb $i | Se Proor.—Will $630, in 9 mo, 18 da,, 
. Tet . — de 
Ans. $630. amount to $665.28, at 7G? . 


Ruie.— Divide the given interest (or amount) by the 
interest (or amount) of $1 for the given time, at the gwen 
rate. 


Prove by trying whether the principal found will produce the given 
interest or amount in the given time, at the given rate. 


852. To find the principal, what must be given? Analyze Example 1. Analyze 
Example 2, Recite the rule for finding the principal, How may the operation be 
proved ? ; : ud : ; 

10 


a 


ZIG INTEREST. 


EXAMPLES FOR PRAOTIOE. 


Prove each example. What principal will yield 

. $91.575 interest, every quarter, at 9%? Ans. $4070. 

. $92.75 interest, in 8 years 6 months, at 5%? Ans. $530. 

$9.75 interest, in 1 yr. 7mo. 6da., at 6%? 

$79.80 interest, at 43%, in 4 years? 

$ .413 interest, at 6%, in 30 days ? 

. $1.65 interest, in 90 days, at 6% ? 

. $40 interest, in 10 days, at 14% a month? 

. What principal will amount to £58, in 2yr., at 8%? Ans. £50. 
9. What principal will amount to $55, in lyr. 8mo., at 6%? 
10. What principal will amount to $12120, in 60 days, at 6%? 
11. What principal will amount to $857.31, in 8mo., at 3%? 
12. What principal will double itself in 10 years, at 10%? 

13. What principal, placed at interest in California at the legal 
rate, on the Ist of September, would amount to $1047.874 on the 


COST & OU HB GO DO bet 


16th of the following January ? Ans. $1010. 
14, What sum lent at 4% a month, Feb. 1, 1865, would amount 
July 22, 1865, to $15427.50 ? Ans. $15000. 


15. How much must a lady invest at 6% in her son’s name, 
when he is just twenty ycars old, that on arriving at twenty-one 
he may have $10000? 

16. How much must a gentleman invest for his daughter at 
7%, that she may have $630 a year? 

17. What investment at 54% will yicld a person a semi-annual 
income of $500? Ans. $20000. 

18, What sum invested at 6% will yield $65 a month? 

19. What sum invested at 7% will yield $3.50 a day ? 

20. A person having $100000 invested at 644%, gives his son 
sufficient to yield a quarterly income of $650, and his niece 80% 
of that amount. How much does he retain ? Ans. $28000. 

21. A’s property, invested at 7%, yields him $8050 a year. 
B’s income is 90% of A’s, and his property is invested at 64. 
Which is worth the most, and how much ? Ans. B, $5750. 

22. Atwhat rate will $200, in 2mo, 12da., produce $1 interest? 


COMPOUND INTEREST. 279 


Compound Enterest. 


353, Compound Interest is that which accrues on inter- 
est due and unpaid, as well as principal. 


Compound interest can not be collected by law, yet it is often 
claimed on the ground that, since the debtor has the use of the interest, 


he should pay for it as well as for that of the principal. 


Savings Banks 


pay compound interest to those who do not draw their interest when it 


is due. 


354, Interest may be compounded annually, semi-an- 
nually, quarterly, or for any other term, according to the 
time at which the interest is originally made payable. 


Ex. 1.—What is the compound interest of $600, at 7 %, 


for 2 yr. ? 


COMPOUNDED ANNUALLY. 

As the interest is to be com- 
pounded annually, we find the 
amount of the principal, at 7 2, 
for 1 yr., by multiplying it by 1.07, 
the amount of $1, at 7 %, for 1 yr., 
being $1.07. This becomes a new 
principal, of which, in like manner, 
we find the amount for the second 
year. This amount, diminished by 
the original principal, will be the 
compound interest required. 

$600 

1.07 

4200 
6000 


642. 
1.07 
4494 
6420 
Amt. for 2yr., 686.94 
Less principal, 600.00 
Compound int., $86.94 


Principal, 


Amt: for 1 yr., 


CoMPOUNDED SEMI-ANNUALLY. 


In compounding semi-annually, 
we find the amount of the princi- 
pal for 6mo., by multiplying it by 
1.035, the amount of $1 for 6 mo. 
being $1.035. Taking this as a 
new principal, we find the amount 
for a second period of 6 mo.; then 
the amount of this result for a third 
period, and of this last amount for a 
fourth—making in all 2yr. We 
then subtract the original principal. 


$600 Principal. 


1.035 
621. Ami. for 6 mo. 
1.035 
642.735 Amt. for 12 mo. 
1.0385 
665.280 Amt. for 18 mo. 
1.085 
688.5138 Amt. for 24 mo. 
600.000 
$88.513 Compeund int. 


858. What is Compound Interest? 


Can it be collected by law? On what 


eround is it claimed? What institutions pay compound interest ?—354. For what 
terms may interest be compounded? Go through the example, explaining the steps. 


220 COMPOUND INTEREST. 


Ex. 2.—F ind the compound interest $686.94 
of $600, for 2 yr. 6mo. 18 da., at 74, 1.0385 
interest payable annually. 343470 

We find the amount for 2 yr., the number of 549552 
entire periods, as above. This we multiply by ee 


1.0385, the amount of $1 for the remaining time, euade: 
6 mo. 18 da., being $1.0385. The product is the $713.387190 
amount at compound interest for 2 yr. 6 mo. 18 600. 
ee re cae 
355, Rute.—Mnd the amount of the given principal 
to the time when the first interest is due. On this amount 
compute the amount for a like period, and so proceed as 
many times as payments of interest are due,—always 
taking the last amount for the new principal. If any 
time then remains, find the amount for such time ; and, 
to obtain the compound interest, subtract the originat 
principal from the last amount. 


EXAMPLIS FOR PRAOTIOE. 


1. What is the compound interest of $1000, for 3 years, at 7%, 
interest payable annually ? Ans. $225.043. 
2. What, if the int. is payable semi-annually? Ans. $229.255. 

3. What is the compound interest of $630, for 4 years, at 5 %, 
interest payable annually ? “Ans. $1835.769. 
4, What, if the int. is payable half-yearly ? Ans. $187.593. 

5. What is the amount of $50, at compound interest for 8 yr., 

at 8%, interest payable yearly ? Ans. $62.985. 
6. What, if the interest is payable quarterly? Ans. $68.412. 

7. Find the compound interest of $800, from Jan. 17, 1862, to 
“April 26, 1866, at 6 Z, interest payable yearly. Ans. $226.646. 
- 8. Find the amount of $740, from Dec. 20, 1868, to Nov. 2, 
1866, at 6%, interest compounded semi-annually. Ans. $876.785. 
9. What will $1700 amount to, in 2yr., at 6%, interest being 
compounded half-yearly? What, if compounded annually? What, 
if compounded quarterly ? Last ans., $1915.04 +. 


Explain Example 2.—855. Reeite the rule for finding compound interest. 


COMPOUND INTEREST. 221 
10. Find the compound int. of $333, at 5%, from May 15, 1863, 
to Nov. 15, 1865, interest payable half-yearly. Ans. $43.75 +. 


3096, The following Table may be used with great ad- 
vantage in calculating compound interest :— 


TABLE, 


Showing the amount of $1 or £1, for any number of years from _ 
1 to 30, at 3, 4, 5, 6, and 74%, interest compounded yearly. 

For the compound interest, subtract 1 from the numbers in . 
the Table. 


Years. | 3 per ct. 4 per ct. 5 per ct. G per ct. 7 per ct. 


1.030000 


1.060900 
1.092727 
1.125509 
1.159274 
1.194052 
1.229874 
1.266770 
1.304773 
1.843916 
1.384234 
1.425761 
1.468534 
1.512590 
1.557967 
1.604706 
1.652848 
1.702438 
1.753506 
| 1.806111 
1.860295 
1.916103 
1.973587 
2.032794 
2.093778 
2.156591 
2.221289 
2.287928 
2.356566 
2.427262 


1.040000 


1.081600 
1.124864 
1.169859 
1.216653 
1.265319 
1.315932 
1.368569 
1.423312 
1.480244 
1.539454 
1.601032 
1.665074 
1.731676 
1.800944 
1.872981 
1.947900 
2.025817 
2.106849 
2.191123 
2.278768 
2.869919 
2.464716 
2.563304 
2.665836 
2.772470 
2.883369 
2.998703 
3.118651 
3.243398 


1.050000 


1.102500 
1.157625 
1.215506 
1.276282 
1.340096 
1.407100 
1.477455 
1.551328 
1.628895 
1.710339 
1:795856 
1.885649 
1.979932 
2.078928 
2.182875 
2.292018 
2.406619 


2.526950. 


2.653298 
2.785963 
2.925261 
3.071524 
3.225100 
3.386355 
3.555673 
3.783456 
3.920129 
4.116136 
4.321942 


1.060000 


1.123600 
1.191016 
1.262477 
1.338226 
1.418519 
1.503630 
1.593848 
1.689479 
1.790848 
1.898299 
2.012197 
2.182928 
2.260904 
2.396558 
2.540352 
2.692773 
2.854339 
3.025600 
8.207135 
8.399564 
3.603537 
3.819750 
4.048935 
4.291871 


- 4.549383 
4.822346 


5.111687 
5.418388 
5.743491 


1.070000 
1.144900 


4+,225043 


1.310796 
1.402552 
1.500730 
1.605781 
1.718186 
1.838459 
1.967151 
2.104852 
2.252192 
2.409845 
2.578534 
2.759032 
2.952164 
3.158815 
3.379932 
3.616528 
8.869684 
4.14.0562 
4.430402 
4.740530 
5.072867 
5.427433 
5.807353 
6.218868 
6.648838 
7.114257 
7.612255 


O02. COMPOUND INTEREST. 


Ex. 11.—Find, by the Table, the ¢ompound interest 
of $400, at 4%, for 12 years, interest due ‘yearly. 
Looking down the column headed 4 per cent., we > 
find the number opposite 12 years to be 1.601032, -601032 
which is the amount of $1 at compound interest, for 400 
the given time, at the given rate. Subtracting 1, we 940.412800 


have .601032 for the compound interest of $1. The 29 
compound interest of $400 is 400 times as much. Ans. $240.41 


Ex. 12.—What is the compound interest of £90, for 
10 yr. 8mo., at 6%, interest payable yearly ? 


£1.790848 
We find, from the Table, the amount of 90 
£1, for 10 yr., at 6%, to be £1.790848. For 161.176320 
£90, it will be 90 times as much. 8 months 1.04. 
remain; find the amount for this time by se 
multiplying by 1.04, the amount of £1 for 644705280 
8 mo being £1.04, Subtracting the origi- 1611763200 
nal principal from the last amount, we get 167.62337280 
for the compound interest £77.6233728,— 90. 


or, reducing the decimal to lower denomina- 


tions, £77 12s. 5d. 2 far. + £77.6233728 = 


£77 12s. 54d. + Ans. 
~ Required the interest, compounded annually, of 
13. $100, for 17 years, at 6%. 

14. $625, for 18 years, at 5%. 

15. $379, for 30 years, at 3 %. 

16. $49, for 20 yr. 2mo., at 64. 

17. $875, for 12 yr. imo. 15 da., at 62%. 
18. What is the compound interest of $100, for 3 yr., at 64%, 


interest payable every six months? 


In this case, the periods are 6 months each. The interest of ¥1, for 6mo., at 
§ per cent., equals the interest of $1, for Lyr., at 8 per cent. There are 6 periods 
of 6mo.,in3yr. Hence we find in the Table the amount opposite to 6 yr. in the § 
per cent. column, subtract 1 since the énterest is required, and multiply the re- 
nuainder by 100, the given principal being $100. 


19. Find the compound interest of $480, from Jan. 1, 1860, te 
July 1, 1862, at 8%, interest payable semi-annually. 

20. What will $1200 amount to in 8yr., at 10%, interest com- 
pounded half-yearly ? 

21. What will $1450 amount to in 10 yr. 6 mo., at 6%, interest 
_ being compounded semi-annually ? we Me 


Ans. $169.277. 
Ans. $1504.137. 
Ans. $919.982. 


v 


oao~= 


i 


- 


. 


NOTES. af: ee 
Vv Ss | 


CHAPTER XXII. 
NOTES.—PARTIAL PAYMENTS:—ANNUAL INTEREST. 


357. A Note (also called a Promissory Note or Note 
of Hand) is a written promise to pay a certain sum to a 
person specified, or to his order, or to the bearer, at a 
time named or on demand. 

358, The Drawer or Maker of a note is the one who 
signs it. The Payee is the one to whom it is made pay- 
able. ‘The Holder is the person who has it in possession. 

The Face of the note is the sum promised. In the 
body of the note the number of dollars is written out, 
and at the top or bottom expressed in figures. 


For example, Jacop Coorer is the drawer of Note 1, given below ; 
Rurus 8. Brown is the payee; the face of the note is $309. 


359, Promissory NotTEs. 
hy d) 
$300. Baltimore, April 9, 1866. 
Sixty days after date, I promise to pay Rufus S. Brown, or 
order, three hundred dollars, value received. 
JACOB CoopER. 
| (2) 
Savannah, Jan. 31, 1866. 
Lor value received, thirteen months after date, we promise to 
pay Messrs. Root & Swan, or order, one hundred and. forty-five 


$25 dollars, with interest. 
Homer F. GREEN. 


$145.50 Moses WATERBURY. 


A note should always contain the words value received. Otherwise, 
if suit is brought on it, the holder may have trouble in proving that the 
drawer received a valuable consideration. 


Note 2 is signed by two parties, and is therefore called a Joint Note. 
It contains the words with interest, and hence carries interest from its 


857. What is a Note ?—858. Who is meant by the Drawer or Maker of a note? 
By the Payee? By the Holder? Whatis the Face of the note ?—359, Learn the 
forms. What words should a note always contain, and why? What is Note 2 
ealled, and why? What is the effect of the werds with intercst? Can interest be 


224 PROMISSORY NOTES. 
aie its 
date, at the legal rate of the State. If these words are omitted, as in Note 


1, no interest can be collected,—unless the note is not paid at the time 
specified, in which case it accrues from that date. 


A bank-bill is a note signed by the president and cashier, payable in 
specie to the bearer on demand,—that is, whenever presented. 

360. A note is said to mature on the day that it be- 
comes legally due. This is not till the third day after 
the time specified in the note, three days of grace, as they 
are called, being allowed, unless the words without grace 
are inserted, If the last day of grace is Sunday or a pub- 
lic holiday, the note matures on the preceding day. 

The term months, used in a note, means calendar months. Thus, Note 
2 is nominally due at the expiration of thirteen calendar months, that is 
on the last day, or 28th, of February, 1867; it is legally due on the third 
day thereafter, March 8d—and interest must be computed for 1 yr. 1 mo. 
3da. It would have matured om the same day, had it been dated Jan. 
30, 29, or 28. 

361, A note to bearer may pass freely from hand to 
hand. <A note to order, to be thus passed, must be signed 
on the back, or endorsed, by the payee. Thus endorsed, 
it is said to be negotiable. 

An endorser is responsible for the payment of the note, if the maker 
fails to meet it at maturity, unless the words without recourse appear 
above his name on the back. If there are several endorsers, the holder 
of the note may look to any or all of them for payment; each is respon- 
sible to those that endorsed after him, and the first endorser has his 
remedy against the drawer, 

To make the endorsers responsible, the holder of the note, if it is not 
paid at maturity, must, on the same day, have it protested by a Notary 
Public, and.serve a notice of protest on each endorser. 

362, A Bond is a written instrument by which a party 
binds himself to pay to another a certain sum, under a 
penalty usually twice the face of the bond. 


collected, if these words are omitted? What is a bank-bill?—860. When is a note 
said to mature? What is meant by days of grace? What does the term months, 
used in a note, mean? Illustrate this in the case of Note 2.—361. How is a note to 
order rendered negotiable? If the maker fails to meet the note at maturity, who is 
responsible ? If there are several endorsers, in what order are they responsible ? 
What must the holder do, to make the endorsers responsible ?—862, What is a Bond? 
—363. If partial payments are made on notes, &c., where are they entered? What 
are they called ?—364. What rule has been adopted by the courts in most of the States 
for finding the balance due? 


PARTIAL PAYMENTS, ye 225 


Partial Payments. 


063. Partial payments, on account, may be made on 
notes, bonds, or other obligations that carry interest. 
They are entered, with their dates, on the back of the in- 
strumeht, and are therefore called Endorsements. 

364; -W hen such payments are made, different methods 
are used for finding the balance due at the time of settle 
ment. The courts in most of the States have adopted the 
rule prescribed by the Supreme Court of the United States. 


Untrep Srates Rue. 


365, According-to the United States method, the ac- 
count is balanced as often as payments are made that 
equal or exceed the interest due. The interest being first 
cancelled, the surplus of the payment goes towards dis- 
charging the principal, subsequent interest being com- 
puted on the balance of principal. No interest is allowed 
on interest; hence the account is not balanced when pay- 
iments less than the interest are made. 

366. Rutu.—Lind the amount of the given principal 
to the time when a payment or payments were made suffi- 
cient to cancel the interest then due, and from this amount 
subtract such payment or payments. Taking the re- 
mainder for a new principal, treat tt like the former one ; 
and so proceed to the time of setilement. 


It can generally be determined mentally whether a payment exceeds 
the interest due. If it is clear that it does not, proceed at onve to the 
next payment.—Follow the forms given under Examples 1, 2. 

(1) $620, Trey, N. Y., Nev. 1, 1862. 

Fér value received, I promise to pay Thomas 
Jones, or order, six hundred and twenty dollars, on de- 
mand, with interest. CuarLes Ban«s, 


Endorsed as follows :—Received, Oct. 6, 1863, $61.07. 


865. According to the U. 8. rule, how often is the account balanced? To what 
is the payment first applied? To what, the surplus? Why is not the account bet 
anced, when payments less than the interest are made ?—866, Reeite the rule. 


_— 


226 , PARTIAL PAYMENTS. 


March 4, 1864, $89.03. Dec. 11,1864, $107.77. July 20, 
1865, $200.50. Settled, Oct. 15, 1865; what was due? 


It is clear that each payment pee the interest due. Hence we 
must compute the amount to the date of each payment. First find the 
intervals of time by subtraction, then the multipliers at 6%. 


Multipliers at 


Yr. mo. da. Intervals. 6 per cent, 

Date of note, 18620 eu. AL 
1st payment, 1863 10: 6 1lmo. 5 da. 0552 
2d payment, 1864 3 4 4mo. 28 da. 0243 
3d payment, 1864.12. Ty 9mo.. 7 da. 0464 
4th payment, 1865". ty "20 Tmo. 9da. 0365 
Date of settlement, 1865 10 15 2mo. 25da, (0145 

Total, 35 mo. 14 da. “1774 


To prove this work, add the intervals, and see whether their sum 
equals the interval from the date of the note to the time of settlement ; 
also, add the multipliers, and see whether their sum corresponds with the 
multiplier that would be obtained from the sum of the intervals. 


Date of settlement, 1865 10 15 2 of 85 mo., .175 
Date of note, 1862; 118041 4 of 14da., .0024 
2 11 14 = 35 mo. 14da. “iTiy 


The multipliers being thus proved correct, we use them in computing 
the several amounts according to the rule, adding 4 as the legal rate for. 
N. Y. is 7%, and carrying the result to three places of decimals, the last 
of which must be increased by 1 when the next figure is 5 or over. 


Face of note, or given principal, . . $620.000 
Interest on the same to Oct. 6, 1863, date of 1st payment, ; 40.386 
Amount due Oct. 6, 1863, : A : : ; : . 660.386 
First payment, . Z : : : : A é : 61.070 
Balance and new principal, =. . 599.316 
Int. on new principal to Mar, 4, 1864, date of 2d payment, é 17.247 
Amount due March 4, 1864, . : . : : . . 616,563 
Second payment, ‘ ‘ : ; ; ‘ : 4 89.030 
Balance and new principal, . . . 527.533 
Int. on new principal to Dec. 11, 1864, date of 3d payment, ; 28.414 
Amount due Dec. 11, 1864, . ; : . 2 ; . 555.947 
Third payment, . ‘ , ; , : . d : 107.770 
Balance and new principal, . . 448,177 
Int. on new principal to July 20, 1865, date of 4th payment, : 19.085 
Amount due July 20, 1865, . ; 5 : : s . 467.262 
Fourth payment, , ‘ : : : : : ; 200.500 
Balance and new principal, . . 266.762 
Int. on new principal to Oct. 15, 1865, date of settlement, ; 4.409 


Balance due at date of settlement, Oct. 15, 1865, . . $271,171 


UNITED STATES RULE. 9% 


(2) $1200. Boston, Jan. 1, 1857. 
On demand, I promise to pay Eli Hart, or 
order, twelve hundred ‘dollars value received, with in- 
terest. SAMUEL Woon WORTH. 
Attest: 


Gro. S. GRAHAM. 


Endorsements :—Received, Feb. 16, 1857, $200. Apl. 16, 
1859, $300. Dec. 24,1859, $25. May 3,1860,$15. Nov. 
3, 1862, $400. What was the balance due Feb. 3, 1864? 


Find the intervals and multipliers as in Ex. 1. It is clear that the 
3d and 4th payments are less than the interest due; hence we pass them 
over, and find the time from the 2d payment to the 5th.—The legal 
rate in Massachusetts is 6%. 


Multipliers at 


YE NO. Cae Intervals, 


6 per cent. 
Date of note, 1867:2+12- 1 
1st payment, 1855 22-516 1mo. 15 da. 0075 
2d payment, 1859 4 16 2yr. 2mo. Oda. 15 
5th payment, 1862 11. 8 3yr. 6mo. 17 da. 212% 
Date of settlement, 1864 2 38 lyr. 8mo. Oda. O75 
Total, Vyr. 1mo. 2da. A254 
Proor. 1864 2 3 4+ of 85 mo., 425 
vical ameee b Mire tof 2da 0004 
Wag tek ye 4254 
Face of note, or given principal, . . $1200.000 
Interest on the same to Feb. 16, 1857, date of Ist payment, . ~ 9.000 
Amount due Feb. 16, 1857, . 1209.000 
First payment, 200.000 
Balance and new oaacipal: : - 1009.000° 
Int. on new principal to April 16, 1859, date of 2d pay ment, 131.170 
Amount due April 16, 1859, . 1140.170 
Second payment, : : 300.000 
Balance and new anceer, ‘ : : ‘ ; . 840.170 
Int. on new principal to Nov. 3, 1862 , date of 5th payment, 178.816 
Amount due Nov. 3, 1862, 2 5 : d . 1018.986 
Third payment (less than euetiaa : : P ~ $25. 
Fourth payment, “ . “ ; ; : 15. 
Fifth payment, : : : ; : F . 400. 
440.000 
Balance and new principal, . . 578.986 
Int. on new principal to Feb. 3, 1864, date of settlement, 43.424 


Balance due at date of settlement, Feb. 3, 1864, 


. $622,410 


PARTIAL PAYMENTS. © 


(3) $1083, 108340  Mitwavgeg, Wis., Dee. 9, 1860. 

On demand, we promise to pay to the or ‘der of Wm. 

_ K. Root one hundred and eight 543, dollars, value received, with 
interest. Brapzsury, Wut, & Co. 


‘Endorsements :—Received, March 8, 1861, $50.04. Dec. 10, © 
1861, $13.19. “May 1, 1863, $50.11. 
_ . low much was due October 9, 1865 ? Ans. $5.844. 


(4) $350. Wimrineron, N. C., May 1, 1862. 
ears For value received, we jointly and severally promise 
to pay Conover, Clark, & Co., or order, on demand, three hundred 
and fifty dollars, with interest. Anson Haieur. 
: Brns. W. Brossom. 


r Endorsements :—Received, Dec. 25, 1862, $50. June 30, 1863, 
$5. Aug. 22, 1864, $15. June 4, 1865, $100. 
What was due on taking up the note, April 5, 1866? 

: Ans. $251.62. 
Pak. A note for $143. 50, dated Aug. 1, 1862, bears the following 
endorsements :—Received, Dec. 17, 1862, $37.40. July 1, 1863, 
ss: $7.09. Dee. 22, 1864, $13.18. Sept. 9, 1865, $50.50. How much 
is due Dee. 28, 1865, the rate being 7% ?: Ans. $60.866. | 

__ 6. On a note for $3240, dated Dec. 1, 1859, at 5 4, the following 
- payments were made :—Dee. 1, 1860, $100. Dec. 1, 1861, $100. 
_‘Dee. 1, 1862, $100. Nov. 1, 1868, $2500. . Oct. 15, 1864, $20, 

ees, ee What was due Aug. 1, 1866? 
Ans. $1177. 244, 

en A note for $486, ¢ dated Sept. 7, 1868, was endorsed as fol- 
lows :—Received, March 22, 1864, $125. Noy. 29, 1864, $150. 
“May 13, 1865, $120. What was the balance due April 19 1866, 

_ the rate being 74? | # . 
--8. A note for $8000, dated June 20, 1858, bor 
_ endorsements : :—Received, Jan. 20, 1861, 000. iM 
$1000. Dec. 5, 1861, $175. , July 31, 186s, $150 
1864, $100. What was the balance due e May: 31, 181 
: ees ga) | s 


MERCANTILE RULE. 


~MmRoanTILE RULE. 


867. Merchants, in computing what is due on accounts 
and notes bearing interest, when partial payments have 
been made, generally strike a balance for successive pe- 
riods of one year, allowing interest on the original princi- 
pal and the several balances, and also on payments made 
‘during each year, from their date to its close. a8 


368, Runzu.—1l. Lind the amount of the given principal 
Sor one year, and from it subtract the amount of each & 
payment made during the year, from its date to the end 
of the year ; the remainder forms a new principal. 

2. Proceed thus for each entire year that follows, to-— 
gether with such portion of a year as may intervene bee 
tween the expiration of the hast @ agree term and thet time. ee 
Of settlement. — ee 


Ex. 9.—According to the mercantile rule, what 4 was. “ 
the balance due ore 1, p. 225, at the Lee of settling ? 


Face of note, or given pri E Noy. 1, 186¢q2 0 $690, 000 ss 
Interest on the saméifor 1 yr., é : : ; . ’ 43.400 
Amount Nove h ISG .e3° 2 ye a a a OCB ee 
Payment made Oct. 6, #863, . -$61.070 | 
Interest on same to Nov. I, 1898, (25 days), : x DOT eee 7 
Balance and new principal, Nov. LAlsesy - ] : S +i ROR08e ~ 
Interest on new principal for 1 yr., . b Aeneas er; S16 we ee 
Agnnush NO tet SOhe. cs Ripa ntl is duct ines BRT 
Payment made March 4, 1864, 7: $89.030 

Int. on same to Nov. 1, 1864, @ mo. 27 da. jae oe 4.103. 


Balance and new principal, Nov. 1, 1864, . ‘ ; Se 
Inteeg date ¢ of settlement, Oct. 15, 1865, 


made Dec. 11, 1864, . $1047.770 
to Oct. 15, 1865, (10 mo. 4 da. ys : 6.370 
made July 20, 1865, . . 200.500 
mn is 15, aay ee 25 da. , eects Bap td, 


ts aaa find the balanee” a when sh oo pune 
6. Recite the mercantile rule. 


2 ees + dome at Te. ee ae 

0 StF ee 4 c te Pee 
Rh Ue Riis 

Weg cate tit “ 


- 230 i PARTIAL PAYMENTS. 


10. According to the mercantile rule, what was the balance 
due Sept. 80, 1865, on a note for $1475, dated June 2, 1864, on 
which was paid, Sept. 17, 1864, $200; Jan. 8, 1865, $300; Aug.” 


2, 1865, $400; interest being allowed at 6%? Ans. $664.285y~ 


ue 11. Solve Ex. 4, p. 228, by the mercantile rule. Ans. $252.123.* © 
ees, 12. Solve Ex. 7, p. 228, by the mercantile rule. Ans. $143.553 


369. When the note is settled within a year, find the 
amount of each payment from its date to the time of set- 
tlement, and subtract their sum from the amount of the 
jace of the note from its date to the time of settlement. 


18. A note for $1000 was given July 18, 1865, at 6 4. 8200 
was paid Sept. 10; $140, Dec. 20; $350, April 21, 1866. What 
was due on taking up the note, June 2, 1866? Ans. $347,428. 

14, Required the balance due Aug. 1, 1866, on a note 
$1380, at 64%, dated Oct. 1, 1865, on wh®li' a payment of $ 
was made Jan. 1, 1866, and a like Be on the Ist day 

- every month ther after. Ans. $1097.166. 

_ 15. A note for $600 is dated Jan. 10, 1865, bearing interest at 
7 q%. Payments of $100 each are made March 15, April 18, Aug. 
1, of the same year. Whatis due Sept. 15,1865? Ans. $321.369. 

Connecticut RuueE. v 

870. By the Connecticut rule, the balance is found 
yearly, when a payment is made within the year; when 
“not, the U. 8. method is followed. 

M871. Ruwe.—t.. Fiadshe balance dua ap ihe.close of 


of the principal for the year, the amount of the payment 
or payments of that year from their date to its end, if 
such payment or payments exceed the interest ; if not, “the 
payment alone, without interest, must be subtracted. 


369. What is the method almost universally used for finding the balance dune on 
notes settled within a year ?—370, By the Ine rule, 2 a often, Soa balance 
found 7—371, Recite the Connecticut rule. a ae te 


4 ‘a < 

2 - ; 

a A 

q 7 | | 

iat 7 Fd i 3 .! x > \ 
7 = k ut 


CONNECTICUT RULE. ” ance “2 a 
2. If no payment has been made within a year, find - 
the amount of the principal to the time er the next pay- 
“ ment, and subtract the payment. 


8. Should the time of settlement not coincide with the 
close of an annual term, compute the last amounts to the 
time of settlement, and not to the close of the 4 year. 


. Ex. 16.—By the Connecticut rule, what was due on 
.. Note 2, p. 227 ? 5 


Face of note, or given principal, Jan. 1, 18517, ee Ans . $1200.000 
Interest on the same for 1 Year, "ge epee es, ye 12000 
Amount, Jan. 1, 1858, ~ : ee ‘ - 1272.000 
Payment, Feb. Te 1857, BUTS eo. - $200.00 eee 
Interest on same to Jan. 1, 1858, (101 m: da.),. 10.50 210.500 
Balance and new principal, Jan. 1, 1858, = ee 2 
_. Interest on new principal to April 16, 1859, a yr 3mo. 15 da. y; 
Amount, April 16,1859, —. os ae ; 3 : 
Payment, April 16, 1859, Ree 6 
Balance and new eeipal. April 16, 859, 5 
Interest on new principal for ‘1 year, : ; ; ke, 
Amount, April 16, 1860, 3 
. Payment, Dec. 24. 1859, (less than interest then due), 
Balance and new principal, April 16, 1860, . : , 
Interest on new principal for 1 year, ; ‘ 
Amount; ApruclG, 1864p. 
Payment, May 3, 1860, i " g1p 000 
Interest on same to April 16, 1861, (11 mo. 13 da. )y _ 858 
Balance and new principal, April 16, 1861, . a 
Interest on new principal to Nov. 8, 1862, (L yr. 6 mo. 17da. ) = 


Amount, Nov. 3, 1862, Pee an Aap gee ‘ 
2 Payment, Noy. 3 , 1862, Rear oye ; : . 400.000 
Balance and new Spepoipal, PAR Ss 589.777 
Interest to date of settlement, Feb. 3, “1864, ‘ : : . | 44,283 


Balance due Feb. 3, 1864, Ore tahoe A ace ae SOseOTR 


aoa to the: Aaa rule, : | 
= SaUE, at was due ¢ on Note 8, p. 228? | Ans. $5.798. 
Find the answer to Ex. 8, p.228. = Ans. $6405.66. 

Vhat is due July 4, 1866, on a note for $9500, at 6 %, dated 
, —$3000 having been paid on account, Aug. 1,1864 
"1865 ; $175, May 30, 18662... --~ains. oe 4: 


AS he NOTES WITH 


Notes with interest anmually. 


372. Notes sometimes contain the words with interest 
annually. In such cases, if the interest is not paid, the 
law in New Hampshire allows the creditor simple inter- 
est on each item of annual interest from the time it ac- 
crued to the date of settlement. 

Ex. 1.—A note for $2000 is given March 17, 1863, with 
interest at 6%, payable annually. No interest having 
been paid, what is due May 3, 1866, according to the law 
of N. H.? 


Face of note, on interest from March 17, 1868, .. ‘ . $2000.000 
Interest on same to date of settlement, May 3, 1866, . ‘ 375.333 

Annual interest, $120, has accrued 3 times. ; 
Interest on $120 from March 17, 1864, to 


date of settlement, ; «2 yr. 1.mo,, Peds. 
Int. on $120 from March 17: 1865, : lyr. 1 mo. 16 da. 
Int. on $120 from March 17, 1866, A 1mo. 16 da. 
Total time, . : ; 3 yr. 4mo, 18 da. 
Interest on $120, for 8 yr. 4mo. 18da.,. : : 5 24.360 
Amount due May 83,1866, . «. . %. | $2899.693 


In stead of computing the interest separately on each item of annual 
interest, it is shorter to add the periods, as above, and find the interest 
on 1 year’s interest for a time equal-to their sum. 

Rviu.— Add the given principal, its interest from date 
to the time of settlement, and the interest on 1 year’s in- 
terest for a term equal to the sum of all the periods during 
which successive payments of interest have been due. Their 
sum is the amount at annual interest. 


This amount will be less than the amount at compound interest, as 
only simple interest on the interest is allowed. 

373. If partial payments have been made on notes 
“with interest annually”, the balance due is found, ac- 
cording to usage in New Hampshire, by the Mercantile 
Rule, § 868, which is therefore sometimes called the New 
Hampshire Rule. 


32. What words do notes sometimes contain? If no interest is paid on such 
_ notes, what does the law in New Hampshire allow the creditor? Go through Ex- 
ample 1, What short method is suggested? Recite the rule for finding the amount 


INTEREST ANNUALLY. Zac 


EXAMPLES FOR PRAOTIOE,. 


1. What amount is due July 5, 1866, on a note for $820, dated 
Jan. 3, 1864, at 5%, interest annually, no interest having been 
paid ? Ans. $926.851. 

2. Find the amount due on a note for $1125, interest payable 
annually at 6%, said note having run 3 yr. 9mo. 9 da. without any 
payment. Ans. $1401.879. 

3. What is due on a note promising to pay $560 five years 
after date without grace, with interest at 53%, payable annually, 


no payment having been made till maturity ? Ans. $730.94. 
4. Required the amount of $290.50, for 6 yr. 2mo., at 6%, in- 
terest payable annually. Ans. $414.718. 


5. Find the amount of $425, for 4 years, at 4%, interest pay- 
able annually. 

6. Required the amount of $850.75, for 8 yr. 10 mo. 6 da., at 
6%, interest payable annually. 

7. A note for $715, dated Dover, N. H., Oct. 4, 1863, bearing 
interest at 6% payable annually, is endorsed as follows: Received, 
April 4, 1864, $75 ; Oct. 1, 1865, $10; Dec. 8, 1865, $100. What 
is due, April 28, 1866? (See § 368.) Ans. $683,257 

Ae 


rae “Ss —-_—-—_—. 
P LASS 


fz a ‘. f ; 1 Nugh 
PP TE NAN, fs SPL AS 
fe in rs VAY : 
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. 


GHAPTER xxi.’ //! 
DISCOUNT. 


374, Discount is an allowance made for the payment 
of money before it is due. | 
070. Discount is often computed without reference to 
time, at a certain per cent. on the amount due, and may 
exceed legal interest. This, however, is not true discount. 


of a note with interest payable annually. How does this. amount compare with the 
amount at compound interest ?—373. If partial payments have been made on notes 
with interest payable annually, how is the balance found in N. H. ? 

874. What is Discount ?—375. How is discount often computed ? 


234 DISCOUNT. 


For example: A merchant buys $1000 worth of goods on 6 months’ 
credit. The money being jvorth more to the seller than its mere interest, 
he will make a discount of 54 on the face of the bill for cash; that is, 
the buyer can discharge his debt of $1000, due in 6 months, by paying 
$950 down. 


Present Worth.—TFrue Discount. 


-376, The Present Worth of a sum due at a future time 
without interest, is such a sum as put at interest for the 
given time will amount to the debt. 

The True Discount is the difference between the present 
worth and the face of the debt. In other words, it is the 
interest on the present worth for the given time. 

if I owe $106 a year hence without ‘interest, and money is bringing 
6 %, the present worth is $100, because that sum at 6 %, for 1 year, would 
amount to $106. The frue discount is $106—$100, or $6; which is the 
interest on $100, at 64, for 1 year. 

377. It will be seen that the debt corresponds to the 
amount, of which the present worth is the principal. 
fence, to obtain the present worth from the debt, the 
rate and time being given, we have only to apply the 
rule in § 352. 


878. Rutn.—1. Zo find the present worth, divide the 
debt by the amount of $1, for the given time, at the given 
rate. 

2. To find the true discount, subtract the present worth 
Srom the debt. 


Exampie.—W hat is the present worth of $124.20, due 
in 6 months without interest, the current rate being 7 % ° 
What is the true discount ? 

Amount of $1, for 6 mo., at 77, $1.035. 


$124.20 — 1.035 = $120, present worth. 


$124.20 — $120 = $4.20, true discount. § Answers. 


Give an example of discount computed without reference to time.—876. What 
is the Present Worth of a sum due at a future time without interest? What is the 
True Discount? Illustrate these definitions.—378. Recite the rule for finding the 
present worth and true discount. Solve the given example. 


EXAMPLES. 235 


EXAMPLES FOR PRAOTICE., 


1. What is the present worth of $4161.575, due three months 


hence, when money brings }% a month ? Ans. $4070. 
2. Of $622.75, due 34 years hence, at 5%? Ans. $530. 
3. What is the true discount on $100, due in 6 months, when 
money is worth 6% ? Ans. $2.9138. 
4, On $750, due 9 months hence, at 7% ? Ans. $37,411. 


5. Find the present worth of $7102.72, due 4 yr. 12 da. hence, 
at 8%. What is the true discount ? 

6. A debt of $150 is due Oct. 1, 1866; what amount would 
pay it, June 13, 1866, reckoning at 64? Ans. $147.847. 

7. Bought, May 1, $50 worth of goods, on 6 months’ credit. 
What sum paid Aug. 1 will discharge the debt, money being worth 
44% per annum ? Ans. $49.443. 

8. A owes B $961.13, due 1 year 5 months hence, and $3471.20, 
due in 8 years 9 months, without interest. Money being worth 
7%, what discount should be allowed on both debts, if paid at 


once ? Ans. $808.448. La 
9. What sum paid down Jan. 1 is equivalent to $37.40 paid on Ap, 


the ist of the next August, money being worth 6%? 

10. A merchant buys a bill of $1500, on 6 months’ credit, but 
settles it by paying cash, a discount of 5% on the face of the bill 
being allowed. What does the discount amount to, and by how 
much does it exceed the true discount, money being worth 7%? 

| Ans. $75; $24.28. 

11. When money brings 4% a month, a merchant settles a bill 

of é 40, due 60 days hence, for cash, at a discount of 24%. What 

does he pay down, how much discount does he get, and by how 

much does it exceed the true discount ? Last ans. $12.68. 

> 12. Sold $1500 worth of goods, on 74 months’ credit. What 
is the present worth of the bill, computed at 74? 

_ 18. Bought, on 6 months’ credit, muslins for $123, hosiery to 

th amount of $100. 50, and $750 worth of cloth. If cash is paid 


| for the wholo bill, what amount should be deducted, reckoning at 


Tow much, at 7%? First ans. $28.35, 


236 ‘ DISCOUNT. 


14. Sold $1500 worth of hardware, half on 6 months’ and half 
on 9 months’ credit. What sum paid down would discharge the 
whole debt, the current rate of interest being 7%? 

15. A man buys a farm of 97 A., at $110 per acre, on a credit 
of 9mo. What discount should be allowed if the money is paid 
down, reckoning at5%? At 64%? Last ans. $495.984. 

16. Bought goods to the amount of $1200, one third payable 
in 8mo., one third in 6mo., and the rest in 9mo. What sum 
paid down would discharge the whole debt, money being worth 
6 per cent? Ans. $1165.21. 

17. Which is worth most, 8500 cash down, $516 six months 
hence, or $530 in twelve months, money being worth 7%? 

18. A merchant, having bought some goods, has his choice 
between paying the face of the bill, $1050, in 90 days, or paying 
cash at a discount of 2%. If money is worth 7%, which had he 
better do, and what will he gain by so doing ? 

Ans. Pay cash; gain, $2.94. 


Bank Discount. ys 

379, A Bank is an institution chartered by law, for 
the purpose of receiving deposits, loaning money, and 
issuing notes, or bills, payable on demand in specie,— 
that is, in gold or silver. 

380. Banks loan money on notes. Deducting a cer- 
tain part of the face of the note in consideration of ad- 
vancing the money, the bank pays over the rest to the 
borrower. The note is then said to be discounted. It 
thus becomes the property of the bank, which, when it 
matures, receives from the drawer the amount of its face. 

The portion deducted, or allowance made to the bank, 
is called the Bank Discount, The sum paid to the holder 
is called the Proceeds or Avails of the note. 

A merchant holds a note for $200, payable in 90 days. ‘Wishing to 


379. What is a Bank ?—880. When is a note said to be discounted? What is 
meant by Bank Discount? What is meant by the Proceeds or Avails of the note? 
Illustrate this process amd these definitions. 


~ 


BANK DISCOUNT. , 237 
use the money immediately, he endorses it, takes it to his bank, and 
places it in the discount box. If both maker and endorser are considered 
responsible, the bank retains the note, and, deducting $3. 10, pays over the 
balance $196.90 to the holder. The Bank Discount is $3.10; the Pro- 
ceeds are $196.90. =e 

381, Bank discount is greater than true discount,— 
the former being computed on the face of the note or 
amount, the latter on the present worth or principal. It 
is equivalent: to simple interest paid in advance, for three 
days more than the time specified in the note,—thr ee days 
of grace being always allowed in computing bank discount. 

382. Casz L—TZo find the bank discount and proceeds 
of a note, rts face being given. 

Ex. 1.—A holds a note for $1000, dated Feb. 1,1866, 
payable in 4mo. April 1, he gets it discounted at 6 
W hat are the bank discount and proceeds ? 

Two months having expired at the date of discount, IntereEt just be 
computed for 2mo. 8 “da. 

Interest of $1 for 2 re ae ee 
$1000 see 7 Peo NO, Proceeds. t Answers. 

Ruize.—1. Hor the bank discount, find the interest on 
the face of the note, at the given rate, jor three days more 
than the specified time. 

. Lor the proceeds, subtract the bank discount from 
the face of the note. 

383. If the note bears interest, cast interest as above 
on the amount due at maturity, in stead of on the face 
of the note. 


Ex. 2.—At 7%, what is the bank discount on a note for 


8600, payable in 6 mo. with interest at 6%? 


Amount of $600, for 6 mo. 8 da., at 6%, $618.30. 
Interest on $618.30, for 6 mo. 3 da., at 7%, $22. 
FER $22, 


881. How does bank discount compare with true discount? Why is bank dis- 
count the greater? To what is it equivalent ?—882. What is Case I.? Explain Ex- 
amplel. Recite the rule—3883. If the note bears interest, how must we proceed? 
Solve Example 2. How does it differ from Ex, 1? 


= 


238 BANK DISCOUNT. 


EXAMPLES FOR PRACTIOER. 


1. What is the bank discount on a note for $1000, for 3 mo., 
at 7%? Ans. $18,083. 
2. Ona note for $150, for 6 mo., at 6%? 
3. On a note for $375, for 3 mo. 9da., at 7%? 
4, On a note for $400, for 9 mo. 27 da., at 64%? 
5. Find the proceeds of a note for $472, nominally due Nov. 
_15, discounted the 15th of the previous January, at 7 per cent. 
: Ans, $444.19. 
6. A note for$1800, payable in 60 days, was discounted at a 
-bank at 6%; how much did the holder receive? Ans. $1781.10. 
7. A merchant gets three notes discounted, the first two at a 
broker’s for 6 %, the third at a bank for 7%. What does he receive 
on_all three, the first being for $837.50 payable in 80 days, the 
Pesoad for $650 in 60 days, the third for $6720 in 90 days? 
Ans. $8074.55. 
8. A farmer buys 43 A. 1 R. of land at $80 an acre. Getting 
a note for $4280.75, payable in 90 days, discounted at a bank at 
6%, he pays for his land out of the proceeds; how much has he 
left ? Ans. $754.40. 
9. A builder buys 23250 ft. of boards, at $80 per M., paying 
the bill with his note at 15 days. The seller gets the ibs dis- 
counted at a bank three days afterwards, at 7%; how much does 
he realize for it ? Ans. $695.47. 
10. What is the difference between the bank discount and the 
true discount on a note dated Feb. 1, 1866, for $400, payable in 
90 days, at 7%? Ans. 85c. 
11. What are the proceeds of a note for $426.10, payable in 
57 days, with interest at 4, discounted at a bank for 6%? ($883) 
. Ans. $426.06. 
12. A owes B for 46 bundles of paper, at £1 10s. aream. He 
pays B the proceeds of a note for £100, payable in 80 days, which 
he gets discounted at a bank for 6%. How much Is he then in 
B’s debt? Ans. £38 11s. 
18. A person having a six-month note for $1200, dated May 


7 


Vv 


BANK DISCOUNT. 239 


2, 1866, on the 1st of June gets it discou ted at a bank for 5 %, 
and invests the proceeds in land at $1 per acre. How much land 
does he buy Ans, 11744 A. 
14. If I get a note for $720, payable 4mo. 15 da. hence, with 
interest at 7%, discounted at 6%, what will the discount be ? 


384, Case IL—TZo find for what sum a note must be 
drawn, for a given time and rate, to yield certain pro- 
ceeds. 

Ex.—For what sum must a note be drawn at 90 days, 
that, when discounted at a bank at 64, it may yield $200 
proceeds ? 

Find the proceeds of $1, for the given time and rate. 

Bank discount on $1 for 90 + 3 days, at 6%, $.0155. 

Proceeds of $1, discounted for 95 days, at 6%, $.9845. 

Since $1 yields $ .9845 proceeds, to yield $200 proceeds will require 
as many times $1 as $ .9845 is contained times in $200, or $203.149. 

Proor.—Bank discount on $203.149, at 6 7%, for 93 days, $3.149. 

$203.149 — $3.149 = $200, Proceeds. 

Ruite.—l. Divide the given proceeds by the proceeds 
of $1 for the gwen time and rate. 

2. Prove by finding whether the proceeds of the result 
equal the given proceeds. 3 


EXAMPLES FOR PRACTIOE. 


1. For what sum must a note be drawn, that, when discounted 


for 8 mo., at 6 %, its proceeds may be $600? Ans. $609.45. 
2, What must be the face of a note, that, when discounted at 
5% for 10 mo., the avails may be $1000 ? Ans. $1043.93. 


3. For what amount must I draw my note at 12 mo., that, 
when discounted at 7%, it may yield $100 ? 

4, For what sum must a note dated May 3, payable Nov. 3, 
be drawn, to yield $365, when discounted at 6%? Ans. $376.48. 

5. A man bought a house for $3287 cash. How large a note, 
payable in 90 days, must he have discounted at 6 4, to realize that 
amount? Ans. $3338.75, 


ts —— | 


834, What is Case U.? Explain and prove the given example ¢ _, 
ff 


240 COMMISSION.—BROK ERAGE, 


6. I had three notes discounted at 64%, for 8 mo., 4mo., and 
6 mo., respectively. The proceeds were $600, $400, and $300. 
’ What was the face of each ? Sum of ans. $1827.26. 
7. A merchant had three six-month notes discounted at 5, 6, 
and 7 %, respectively. The proceeds of each were $1000. What 
was the face of each ? First ans. $1026.08. 


CHAPTER, X LILI. 
COMMISSION.—BROKERAGE.—STOCKS. 


385. Commission is a percentage allowed to an agent 
for the purchase or sale of property, the collection or in- 
vestment of money, or the transaction of other business. 

A party attending to such business for a commission 
is called an Agent, a Factor, Commission-merchant, or 
Broker. 

386. A Broker is one who buys or sells goods for 
another, without having them at any time in his posses- 
sion, or who exchanges money, obtains loans, or deals 
in stocks. The commission paid to a Broker is called 
Brokerage. 


The rate of commission and brokerage differs according to the busi- 
ness transacted and the amount involved, ranging from + to 5¢. A com- 
mission-merchant usually gets 24¢-for selling goods, and an additional 
242 if he guarantees the payment. 


387. A Consignment is a lot of goods sent by one 
party to another for sale. The party sending them is 
called the Consignor; the one receiving them, the Con- 
signee. 3 

The Gross Proceeds of a consignment are the whole 


885. What is Commission? Whatisa party attending to business on commis- 
sion called 7—886. What isa Broker? What is Brokcrage? Between what limits 
does the rate of commission and brokerage generally range? What does a commis- 
sion-merchant usually get for selling goods ?—387. What is a Consignment? Who 
45 the Consignor? Who is the Consignee? Whatis meant by the Gross Proeeeds 


Ie. sa fe _- — ‘i_: ts A Py. Ds | Fa Or om (ae 2 oben 2 ae, 7 “ale eh Te 
fe ee ae ee fn Oe eer eee ee MLO xr en A reece ORR, ee eee 
— rq . , wh 


a - : 7 we aad % ai a ae A Pans 


? 


STOCKS. 241 


amount realized by the sale. The Net Proceeds are what 
is left for the owner, after deducting commission and other 
charges. 


388, Stocks is a general term applied to Government 
or State bonds, and the capital of companies incorporated 
or chartered by law. There are state stocks, bank stocks, 
railroad stocks, &c. 


When a company is formed for building a railroad, constructing a 
telegraph line, establishing a bank, carrying on extensive manufacturing 
operations, or any other enterprise, those interested subscribe a certain 
amount needed for conducting the business, which constitutes the Capi- 
tal, or Stock, of the company. This stock is divided into portions called 
Shares, which may be of any amount, but are usually $100 each, and 
are represented by Certificates or Scrip. —Stock is bought and sold by 
brokers. It is constantly fluctuating in value, rising or falling according 
to the demand for it, the profits of the company, and other influences. 

Those who own any particular stock, whether by original subscrip- 
tion or purchase, are called Stockholders. They constitute the Com- 
pany, and elect Directors, by whom a President and other officers are 
chosen. 

389, A broker who deals in stocks is called a Stock- 
broker. His commission for buying or selling is reck- 
oned at a certain per cent. (usually $%) on the nominal 


value of the stock, without reference to the market price. 


390. Commission is a percentage. 

The money collected, realized, or invested, is the dase. 
The per cent. allowed as commission is the rate. 
Hence, by the principles of Percentage (§ 321), these 


Roies.—I. Zo find the commission, multiply the base 
by the rate. 

Il. Zo jind the rate, divide the commission by the base. 

Ill. Zo find the base, divide the commission by the 
rate. 


efaconsignment? By the Net Proceeds ?—3888. What is meant by Stocks? When 


a company is formed, how is the necessary capital obtained? How is this capital, 


or stock, divided? By whom is stock bought and sold? What makes it fluctuate 
in value? Who are called Stockholders? Whom do they elect? Who are chosen 
by the directors ?—389. What is a broker who deals in stocks called? How is a 
stock-broker’s commission reckoned ?—390. Commission being a percenta age, what 
is the base? Whatis the rate? Recite the rules. oh 


IL 


242 COMMISSION. 


EXAMPLES FOR PRAOTIOE. 


[In all the examples relating to stocks, take $100 for a share, un 
less otherwise directed. | 

1. What commission must be paid an agent for collecting bills 
to the amount of $2460, at 5%? Ans. $123. 

2. A broker buys for me 100 shares of Erie R. R. stock, and 
sells the same the next day. What is his brokerage, 4% being 
charged for each transaction ? 

8. A lady, having $22000 on bond and mortgage at 6%, em- 
ploys an agent to collect 1 year’s interest and invest it. What 
commission must she pay, the rate being 244% for collecting and 
44 for investing ? ‘ Ans. $39.60. 

4, What brokerage must a person pay to have $1475 uncur- 
rent money exchanged, at an average rate of 3%, and how much 
should he receive in current funds ? 

5. An auctioneer, who charges 24%, receives $225 for selling 
some paintings; how much did they sell for ? Ans. $11250. 

6. What are the net proceeds of a consignment sold for $4250, 
on which there are charges of $27 cartage, $103 storage, and 24% 
commission ? Ans. $4018.75. A 

7. Sold 412 bales of cotton, averaging 405 Ib. each, @ 27c. a Ib. 
What was the commission, at 24% ? Ans. $1126.305. 

8. What % does a commission-merchant charge, who receives 
$223 for buying $5575 worth of goods ? Ans. 4%. 

9, A factor in Mobile received from a planter 514 bales of cot- 
ton; after paying on it $840 expenses, he sold it at $120 a bale. 
He then bought for the planter $1525 worth of hardware, and 
groceries to the amount of $3018.20. His commission being 2% 
on sales, and 8% on purchases, how much must he remit to the 
planter ? Ans. $54926.90. 

10. An agent collects for a society 250 bills, of $6 eack. How 
much must he pay over, if he gets 5 4 commission ? 

II. A broker sells for a customer 250 shares of N. Y. Central 
R. R. stock, and buys for him 800 shares of Michigan Southern. 
At 4%, what is the brokerage ? 


7 ee oat 


COMMISSION. 243- 


12, Wishing to buy 85 A. of land, I obtained the necessary 
amount through a broker, who charged 14% for negotiating the 
loan. His commission amounted to $63.75; what did the land 
cost per acre ? Ans. $75. 

13. A commission-merchant, nee selling $12000 worth of grain 
and guaranteeing payment, charged $600, and for purchasing a 
bill of $4220 charged $63.30. What % did he charge for each 
transaction ? 

14, A factor, having sold 1250 barrels of flour at $8 a barrel, 
invested his commission, which was at the rate of 12%, in a new 
company that was forming. How many shares; at $25 each, did 
he take? Ans. 7. 


- f 


391. Zo find the base, the rate and the sum or differ- 
ence of the commission and base being given. 


A party sometimes remits to an agent a certain amount 
to be invested, after deducting his commission. Here the 
sum of the commission and base is given, and the base, or 
amount invested, is required. 

Again, when the net proceeds and rate are known, it 
is sometimes required to find the gross proceeds of a sale. 
Here the difference between the base and the commission 
is given, and the base is required. 

These cases are analogous to those presented in § 323, 
under Percentage. 


Ex. 1.—B, sends a commission-merchant $6000 to invest 
in cotton, after deducting his commission of 2%. How 
much must be invested, and what is the commission ? 


Every $1 invested will cost B $1 + 2c. commission, or $1.02. Hence 
there wili be as many times $1 invested as $1.02 is contained times in 
$6000. $6000 + 1.02 = $5882.35, Amount invested. 

The commission will be the difference between the whole amount sent 
and the sum invested. $6000 — $5882.35 = $117.65, Commission. 

Prove by finding whether the commission on 85889. 35, at 24, is 
$117.65, 


891. What cases are sometimes presented, analogous ie those in § 828, under 


Percentage? Explain Example 1. 


& 


244 COMMISSION. 


Ex. 2.—A real estate agent, having sold a house, pays 
himself 1% commission, and hands over to the owner 
$13365. What did the property bring, and what is the 
commission ? 


The commission being 1 4, every $1 of the purchase price will net the 
owner 99c. The house, therefore, brought as many times $1, as 99c., the 
net proceeds of $1, is contained times in $18365, the net proceeds the 
owner received. $13365 + .99 = $138500, Selling price. 

The commission will be the difference between the selling price and 
the net proceeds. $18500 — $13365 = $185, Commission. 


Prove by finding whether the commission on $138500, at 1 Z, is $135. 


392. Rute.—1. For the base, divide the given number 
of dollars by 1 increased or diminished by the rate ex- 
pressed decimally, according as the sum or difference of 
the commission and base is given. 

2. For the commission, take the difference between the 
base and the given number of dollars. 

3. Prove by finding whether the commission obtained 
by multiplying the base by the given rate, equals the com- 
mission as just found. 


EXAMPLES FOR PRACTICE. 


1. A broker receives $30000 to invest in real estate, after de- 
ducting his brokerage of }%. What will be the amount invested, 
and. what his commission ? First ans. $29925.19. 

2. A person sends his commission-merchant $15000 to invest 
in corn. The commission, 1%, being taken out of the sum sent, 
and the corn costing 75c. a bushel, how ‘many bushels were pur- 
chased ? Ans. 19801 bu. + 

3. An agent, having sold some property, paid the owner 
$11137.50, which remained after deducting his commission of 1%. 
What did the property sell for ? Ans. $11250. 

4, A commission-merchant paid $1000.50 charges on a con- 
signinent, retained 24% commission, and remitted $38487 to the 
owner; what were the gross proceeds? Ans. $40500. 


Explain Example 2.—392. Recite the rule for finding the base and commission, 
when the sum or difference of the commission and base is given, 


ACCOUNT OF SALES. 245 


5. An agent, who gets 54%, collects a number of bills of $16 
each, for a society. He pays over to the treasurer $1149.50; how 
many bills were collected ? Ans. 121, 

6. What are the gross proceeds of a consignment, if the com- 
mission is 24%, the other charges are $1000.85, and the net pro- 
ceeds $12772 2 

7. A gentleman who has $30000 invested on bond and mort- 
gage at 7%, employs an agent to collect six months’ interest, and 
directs him to invest in grain what is left after paying himself his 
commission,—which was 1% on the amount collected, and 2% on 
the amount invested. How much was invested? Ans. $1019.12. 


Account of Sales. 


393. An Account of Sales is a statement rendered by a 
commission-merchant to a consignor, setting forth the 
prices obtained for the goods sent and the amount real- 
ized, the charges paid and the net proceeds due the con. 
‘ signor. They are made out in the following form :— 


Sales of Flour for acct. of Rk. Day & Co., Buffalo. 


1866. | Sold to Description. Bar. @ 
June 1 | Beck & Co. | Extra Ohio. 83 | $9.00 | $747.00 
“ 9 | I. R. Shaw. || Canadian. 20 | 9.20 184.00 
“64 | S. Bennett, Phenix Mills. | 95 | 8.75 831.25 
cs Dk. David Orr. eee Gee. OFLU. 652.50 
“© | Roe & Son. oxty Mills. 160 | 7.95 | 1272.00 
$3686.75 
CHARGES. 
Freight on 483 bar., @ Wes $324.78 
Cartage, . ; tite e bo 
Storage, . . 43.30 
Commission on $3686. 75, @ 24%, 92.17% 
Total charges, $481.72 
Net proceeds to credit of R. Day & Co., Are 03 | 
EV &si OnE 5.0 LPAAMAL ay arr" 
N. Y., June 6, 1866. ‘Burrerwortn, Hupson & Co. 


* Errors and omissions excepted. 


. ih, 


G 


yA 


246 ACCOUNT OF SALES. 


1. Make out the following Account of Sales, and find the net 
proceeds due the consignor :— 


Sales of 4265 Bushels Wheat, for acct. of Asa F- Mac 


Onerge. 
1866. Sold to | Description. Bion a vil 
July 2 | H. Brown. | Red Winter. 750 | $1.90 | $ 
pa ant avy cNEtLiss 600 | 1.89 
““ 38 | Bruce & Co. % 3 600 | 1.92 
¢ “| Farr Bro’s. | New Mich. 500 | 2.62 
Bere Le hes hOOs h 2% 940 | 2.61 
eee tts soo E Ute °° Re 875 | 2.59 | 
i § 
CHARGES. 
Freight on 1950 bu., @ 12¢, $ 
“9315 bu, @ 124, 
Advertising, . ‘ ‘ - $5.45 
Commission on $ , at 24%, 
Total charges, $ 
Net proceeds to credit of Asa F. White, $ 
E. & O. E. 
Piet. July 1, 1800. Harrison & Barrow. 


2. Make out an Account of Sales, in proper form, from the fol- 
lowing data :— 

Messrs. Meyer & Herzog, commission-merchants, of New Yerk, 
received a consignment of provisions from Henry L. Jones & Co., 
of Rome, N. Y., as follows :—10 firkins of butter, 940 1b. ; 37 cwt. 
of cheese; 80 barrels mess pork; 16 cwt. hams; 40 packages 
shoulders, 2700 Ib. 

They paid charges on the consignment as follows :—Freight, 
$75.40; drayage, $5.75; storage, $12.25; insurance, $6.50; ad- 
vertising, $12.75. Their commission was 24%. 

They sold the butter, June 19, 1866, @ 374c. alb.; the cheese, 
same date, @19c. June 20, they sold the pork @ $31 per bar., 
the hams @ 183c. a lb., the shoulders @ 14c. 

Ans. Net proceeds, $2484.26. 


yp 


STOCKS :—-TECHNICAL TERMS, 247 


Stocks. 


394, The Market Value of a stock is what it sells for. 


395. When the market value of a stock is the same as 
its nominal value, it is said to be at par. 

When its market value is greater than its nominal value, 
it is said to be above par or at a premium; and when 
less, to be below par or at a discount. 


When a hundred-dollar share sells for $100, the stock is at par; at 
S101, it is above par, or at a premium of 1%; at $99, it is below par, or 
at a discount of 1%. The premium or discount is always reckoned on the 
par value as a base.—Stock is generally quoted at the market value of one ~ 
share. In the three cases just specified, it would be quoted respectively 
at 100, 101, and 99. 

396. When the capital for a new company has been 
subscribed, if it is not all needed immediately it is called 
for in portions, or Instalments—a certain per cent. at a 
time. 

397. Stockholders are sometimes called on to meet 
losses or make up deficiencies, by paying a certain 
amount on each share they hold. The term Assessment 
is applied to a sum thus called for. 


398. The Gross Earnings of a company consist of all 
the moneys received in the course of their business. The 
Net Earnings are what is left after deducting expenses. 
When there are net earnings to any considerable amount, 
they are divided, in whole or in part, among the stock- 
holders, according to their respective amounts of stock. 

399, A Dividend is a sum paid from the earnings of a 
company to its stockholders. 


Let the capital of a company be $2500000 ; let its gross earnings for 


394. What is meant by the Market Valuc of a stock ?—395. When is a stock 
said to be at par? When, above par? . When, below par? When, at a@ pre- 
mium? When, at a discount? Illustrate these definitions.—396. What is meant 
by Instalments ?—397. What is meant by an Assessment ?—398, What is meant by 
the Gross Earnings of a company? By the Net Earnings? "When there are net 
earnings to any considerable amount, what is done with them?—399. What is a 
Dividend ? 


oe Pp a aw 
\ Me 


eas 
e 


7 


248 “stocks. —BONDS 


six months be $250000, and its expenses for the same time $100000: 
the net earnings will be $150000. Of this sum it is thought best to re 
tain $50000 as a surplus, to meet any unforeseen expenses, and to divide 
the rest, $100000, among the stockholders. To find the rate, the per- 
centage must be divided by the base, $321. $100000 + 2500000 = .04. 
A dividend of 4 per cent. is declared ; and each stockholder’s dividend 
will be found by taking 4 % of the par value of his stock. 


490. When a company need money, they sometimes 
borrow it on their property as security, issuing Bonds, 
which bear a certain fixed rate of interest without refer- 
ence to the profits. ‘The income from the stock, on the 
other hand, depends on the net earnings,—the interest on 
the Bonds, as well as other expenses, having been first 
paid. 

401. Cities, counties, and states, may also issue Bonds 
to raise money. ‘These Bonds are named according to 
the interest they bear. Thus, Tennessee 6’s are Bonds 
bearing 6 per cent., issued by the state of Tennessee. 


402. The United States Government has issued sev- 
eral different classes of Bonds and Treasury Notes, which 
constitute what are called “U.S8. Securities” or “ Fed- 
eral Securities”. 


U.S. 5’s of °71 and ’74 are bonds payable respectively in 1871 and 
1874, bearing interest at 54 in gold. 

U.S. 6's of °67, 68, and ’81, are bonds payable respectively in 1867, 
1868, and 1881, bearing interest at 6 4 in gold. 

5-20’s are bonds bearing interest at 6% in gold, so called from their 
being payable in not less than 5 or more than 20 years from their date, 
at the pleasure of the Government. 

10-40’s are bonds bearing interest at 5 4 in gold, so called from their 
being payable in not less than 10 or more than 40 years from their date, 
at the pleasure of the Government. 

7-30’s or 7 8-10’s [Seven-thirties or seven and three-tenths| are 
treasury notes payable in three years from their date; they are so called 
from their bearing interest at 74%5 % in currency, or lawful money. 


403, In the case of sales, brokers have to use a revenue stamp eqval 
in value to 1 cent on each $100 (or fraction of $100) of the currency value 
of the stocks or bonds sold ; this is charged to the parties for whom they sell. 


‘  Tllustrate the mode of finding the rate of a dividend to be declared. How is 


each stockholder’s dividend found ?—400. How is money sometimes raised by a 
company? How does the income from bonds differ from that arising from stock ?— 
401. What besides companies may issue bonds? How are these bonds named? 
Give an example,—402. Name the several classes of United States Securities, 


STOCKS. forse ObDr 8 249 


EXAMPLES FOR PRAOTIOE. 


[Unless otherwise directed, take $100 for a share, and }¢@ for the 
vate when brokerage is paid. ] 
1. What is the market value of 200 shares of N. Y. Central 
R. R. stock, at 97 ? 
If 1 share is worth $97, 200 shares are worth 200 times $97. 
2. What will I have to pay for 200 shares of N. Y. Central, at 
97, and brokerage on the same ? 


1 share will cost $97 + 4 per cent. of $100 (brokerage), or $97.25. 
200 shares will cost 200 times $97.25. 


3. What will I realize on 200 shares of N. Y. Central sold at 
97, over and above brokerage and cost of revenue stamp ? 
200 shares, at 97, $19400.00. 
Deduct brokerage, 1 per cent. on $20000, $50.00. 


“ for stamp, 1c, on 194 hundred dollars, 1.94. 
$50 + $1.94 = $51.94. $19400 — $51.94 = 19348.06. Ans. 


4, What is the market value of 100 shares of Michigan Cen- 
tral, at a premium of 34%? Ans. $10350. 
5. What will 125 shares of Western Union Telegraph stock 
cost, at 80% discount, with brokerage ? Ans, $8781.25. 
6. What will be realized, over and above brokerage and cost 
of revenue stamp, on 500 shares, of $25 each, sold at a premium 


of 24%? Ans. $12748.72. 
7. Bought through a broker 100 shares of Alton and Terre 
Haute at 312; what do they cost? Ans. $3212.50. 


8. Sold Virginia 6’s to the amount of $20000, at a discount of 
30%; and 2000 three-dollar shares of a petroleum stock, at 45 % 
discount. No brokerage being paid, how much is realized from 
the sale ? Ans. $17800. 

9. Bought 50 shares of Ocean Bank stock at par, and sold 
them at 105. What is the profit, brokerage being paid on each 
transaction, and the cost of revenue stamp being deducted ? 

Find the profit on 1 share, by deducting 50c. brokerage from 5, the advance in 
price. Multiply the profit on 1 share by the number of shares, and from the prod- 
uct subtract the cost of stamp. 

10. What is the loss on 250 fifty-dollar shares, bought for 102 
sad sold at 993, taking brokerage and cost of stamp into account? 


Frits 


ead 


eal 


STOCKS. 


11. I buy through a broker 175 shares of bank stoald at 97%, 
and sell them through the same at a premium of 4%; what is my 
profit ? . 

12. Ifa person buys 40 fifty-dollar shares at 18 4 above par, and 
sells them at 114% below par, does he make or lose, and how 
much ? 4 

13. A person exchanges 150 shares of Erie at 60, for stock be 


a Quicksilver Co. at 25% premium. How many shares should he 


- receive ? Ans. 72 shares. 


14. Bought some stock at 92, sold it at 944. Brokerage was 
paid on each transaction. The profit being bie ees many 
shares were there ?. 


Brokerage on 1 share $.50, Cost of stamp on I share sold at 943, $.00945. $.50 + 
$ .00945 = $ .50945. Profit on 1 share, $2.50 — $.50945 = $1. 99055. As many shares 
were solid as $1.99055 is contained times in $398.11. -- 


15. How much stock, at 10% discount, can be bought for 
$4500, brokerage being left out of account ? Ans. 50 shares. 


What will 1 share cost, at 10 per cent. discount ? 
How many shares, at that price, can be bought for $4500? 


_ 16. How much stock, at a premium of 344%, can be bought for - 
$10350, brokerage being paid ? Ans. 100 shares. 

17. A merchant wishes to sell sufficient stock to realize $15000. 
The stock being at 754, and brokerage } 4, how many shares must 
he sell? 

18. Bought 100 shares of Nassau Bank stock at 105. ‘They 
were sold at a profit of $350, leaving brokerage out of account ; 
what premium did they bring? \f Ans. 84% 

19. A broker receives $19100 to Mavest in Kentucky 6’s, bro- 
kerage to be paid out of the amount sent. The stock stands at 


954; how many thousand-dollar bonds can he buy, and what in- 


come will be received from them every year ? 

20. A company with a capital of $750000, having earned 
$22500, put aside $3750 as a surplus. What per cent. dividend 
can they declare? (See § 899.) Ans. 24 %. 

21. In the above company, A holds 10000 worth of stock ; 

B, $20000; ©, $17500. What will their dividends respectively 
Picant to? Ans, A’s, $250, &e. 


: re \ 
Z2"ou\ 


| | 


a =e EXAMPLES FOR PRACTICE. 251 


22. A railroad company having declared a dividend of 3 %, 
tow much will a person who holds 400 fifty-dollar shares receive ? 
23. A mining company, whose shares at par are $25, declare a 
dividend of 1% every month. How much will a party who holds 


- > 1000 shares receive in one year ? 


ol 


24. I hold $5000 worth of 6% bonds im a certain company, and 
50 shares of the capital stock. The company declare semi-annual 
dividends of 34%. What ismy yearly income from both ? 


In these Examples, the government tax of five per cent. on dividends and in- 
terest accruing on all bonds (except those of the U. 8.) is left out of account. 


25. D bought 100 shares of stock at 84, and sold them at 87, re- 
ceiving meanwhile a dividend of 3%. What was his profit? 

26. A company with a capital of $10000000 have $200000 net 
earnings; what dividend can they declare? What dividend will | 
a party receive who holds $10000 of their stock ? 

27. How much stock in the above company does a party hold, 
who receives adividend of $10000 ? 


~ Dividend = Stock x Rate. : 
Hence, Stock = Dividend + Rate. Rate = Dividend + Stock. 


28. What @ dividend does a person get, who receives $350 and 
owns 50 shares of stock ? . 
29. When gold is at a premium of 294, what is $1000 in gold 
worth in currency ? . 
$1 gold = $1.29 currency. $1000 gold = $1.29 x 1000 currency. 
The banks having suspended specie payments in 1861, gold and silver have 


since that time commanded a premium; that is, $1 in gold or silver has been worth 
more than $1 in currency. 


30. When gold is at 129, how much gold will $1290 in cur- 
rency buy ? Ans. $1290 + $1.29 = $1000. 

31. When gold is at 141, how much in current funds will 
$12000 in gold cost? 

32. When gold is at a premium of 254%, how much gold will 
ee in currency buy ? . 

. A lady holds $8000 worth of U.S. 5-20’s; what will she 
receive annually from these bonds in currency, if gold commands 
a premium of 80%? (See § 402.) 

$8000 x .0G = $480 in gold, $450 x 1.30 = $624 in currency, 


~ ¥ ~ a Ds 


ee) 


252 STOCKS. 


34, What is the semi-annual income in currency from $15000 
~_.. worth of U. §. 5-20’s, when gold brings 133 ? 
35. What is the yearly income in currency from $10000 in 
U.S. 10-40’s, when gold is worth 126? Ans. $630. 
36. What is the yearly income from $20000 in U. S. 7-30’s? 
37. What yearly income will one who subscribes for $10000 
of a seven per cent. loan, at par, receive from it ? 
ae ND 38. Ifa person invests $8245 in 6% bonds, at 97, what will be 
_ A ~ his annual income from the investment ? 


rd Each dollar of stock bought costs 97c. Hence, for $8245 can be bought as many 
fi dollars of stock as 97c. is contained times in $8245. Then find the interest on the 


amount bought, at 6 per cent. 
ra 89. What income will be annually received from certain 7% 


rie / bonds, bought at 108, and costing $14420 ? Ans. $980. 

d ‘a A 40. A person invests $19600 in 10-40’s, at 98. What income 
A ~~ in currency will he annually receive from the bonds purchased, 
if gold sells at 1402 ~ Ans. $1400. 


ge 41. When gold is worth 129, what half-yearly income in cur- 
~—=—~rent funds will a person receive.wh Anyests $7540 in U.S. 5-20's, 
Ans. $280. BIT. 


mn then selling at 104 ? 
y 42. When Missour What,sum must be invested 
_ in them, to yield an annuaFigeome of $2700 2 
: te Stock required = Income + Rate. $2700 + .06 = $45000. eo | 
/ 4% 4. — $45000 stock, at 75, will cost $38750 Ans. Hence the following rule :— vA 
aR: £0 4404, Rutz.— To find um must be invested in 


, . bonds, selling at a given raté to secure a given mcome, 
~ <4 4, Find the par value of the stock required, by dividing 
l the given annual income by the annual income of $1 of 
the stock. 

2. Multiply this par value by the market value of $1 
of the stock. 

43. How much must one inyest in Brooklyn 6’s, at 90, to _ 
secure an annual income of $1500 ? Ans. $22500. ~ 

A4, 1f I scll $10000 U. S. 6’s, at 107, and with part of the pro- 
ceeds buy N. Y. Central 6's, at 90, sufficient to yield $300 an- 
nually, how much will I have left ? Ans, $6200. 


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EXAMPLES FOR PRACTICE. 253 


45, When U. S. 7-80’s are selling at 103, what sum must be 
invested in them to yield $1460 a year? What sum invested in 
them will yield a semi-annual income of $109.50 ? 

46. When N, Carolina 6’s are 15% below par, what will be 
the cost of bonds sufficient to yield $1200 yearly ? 

47. Holding a large amount of Erie R. stock, I wish to sell 
part of it and buy Tennessee 6’s sufficient to yield me $1800 a 


year. Erie standing at 60, and Tennessee 6’s at 90, how many ~~ 


shares of Erie must I sell to make the change, leaving brokerage 
out of account ? Ans. 450 shares. 

48. What 4 income will a person realize on his investment, 
who buys 6 per cent. bonds at 96? 

$1 of the stock yields 6c. and costs 96c. The question therefore becomes, What 
per cent. is 6c. of 96c.? Divide the percentage by the base, § 321. 

-06 +- .96 = .0625. Ans. 64 per cent. Hence the following rule :— 

405. Rutz.—TZo find what % annual income is real- 
ized on an investment in stocks at a given price, 

Divide the annual income of $1 by the cost of $1 of 
the stock. 

49. What % income will be realized on 7% bonds bought at 
91? At98? At105? First ans. 75%. 

50. If I get an annual dividend of 74 on stock that cost me 
70, what % do I receive on my investment ? 

51. What % on the investment will a stock bought at 90 yield, 


if a dividend of 3% is paid every six months? _ Ans. 63. 
52. What % on his investment will a person receive, who buys 
U. 8. seven-thirties at 104? Ans. Ti5 % 


53. What % on his investment will a person receive, who buys 

U. S. 6’s at 107, when gold stands at 150? 
206) x5 D0.== 09 .09 + 1.07 = 84% per cent. Ans. 

54, When U.8. 10-40’s are at 97, and gold is worth 125, what 
per cent. will an investment in these bonds yield ? 
_ 55. A person desiring to make a permanent investment, hesi- 
tates between buying U. 8. 7-30’s at 103 and Kentucky 6’s at 95. 
Which will pay him the better 4 on his investment, and how 
much ? - Ans. Seven-thirties, 1549 4, 


254 STOCKS. 


56. Which investment will pay the better Z—and how much— 
5-20’s at 1044, or 10-40’s at 9732 

57. A person having his money invested on bond and mort- 
gage, at 6%, calls it in, and buys Michigan Central 8's, at 110. 
How does his rate of income on the latter investment compare 
with what it was before ? Ans. 1,2,% better. 

58. Which is the best for permanent investment—bd’s at 75, 
6’s at 85, or 7’s at par? 

59. A party investing in 5 per cent. bonds realizes 8% income 
on his investment. How did the bonds stand when he bought? 


$1 of the bonds yields 5c. The question therefore becomes, 5c. is 8 per cent. 
of what? Divide the percentage by the rate, §231: 
.05 + .08 = .625, cost of $1 of the bonds. 
.625 x 100 = 62.5, cost of $100 of the bonds. Ans. 62}. 


60. What must one buy a 74% stock for, to realize an income 
of 8% on his investment ? Ans. 874. 

61. How much above par does an 8% stock sell for, when it 
pays an interest of 7% on the investment? What must it sell for, 
to pay an interest of 9% on the investment ? 

62. When gold stands at 130, what must a party buy 5-20’s 
for, to realize 7% on his investment ? 


$1 of the bonds yields $.06 in gold, or (.06 x 1.30) $.078 in currency, Then pro- 
ceed a8 in Example 59. i 


63. When gold is at 135, what must eae for, to yield 
8% interest on the investment ? Ans. 84%. 

64. What must gold sell for, that a party investing in 5-20’s, 
at 105, may realize 8% interest. on his investment? — 


$1 of 5-20's yiclds $ .06 in gold, and costs $1.05. 
Hence, § 231, .06 + 1.05 = .053. The interest on the investment, in gold, is 
therefore 53; and, to pay 8 per cent. in currency, gold must sell for as much as 53 is 
contained times in S, or 1.40. Ans, 40 per cent. premium, or 140. 


65. What must gold sell for, that an investment in 10-40’s at 
97 may yield an interest of 7%? Ans. 1354. 
66. Which is the better investment, U. S. 5-20’s at 104, gold 
standing at 125, or Virginia 6’s at 70—and how much? 
67. If I sell 200 shares of stock at 49, paying brokerage, and 
invest the proceeds in 10-40’s at 974, what will be my annual in- 
come when gold is 130? ; Ans. $650. 


BANKRUPTCYe 255 


CHAPTER XXIV. 
BANKRUPTCY. 


406. A Bankrupt is one who fails in business, or is 
unable to meet his obligations. Such a party is said to 
be insolvent. 

The Assets of a bankrupt are the property in his hands. 
His Liabilities are his debts, or obligations. 

407, When a person becomes bankrupt, an Assignee 
is usually appointed, who takes possession of the assets, 
turns them into cash, and, after deducting his own 
charges, divides the net proceeds among the creditors 
in proportion to their claims. 

ExampLe.—A merchant fails, owing A $3000, B $6250, 
C $800, and D $9950. His assets are $8650, and the ex- 
penses of settling $650. What can he pay on the dollar, 
and how much will each creditor receive ? 


We must first find the rate of dividend. The total of liabilities is the 
dase; the net proceeds of the assets, the percentage. Dividing the per- 
centage by the base, § 321, we find the rate to be 404%, or 40 cents on 
the dollar. Each creditor’s share is then found by multiplying his claim 
by this rate. 

Prove by finding whether the sum of the several dividends corresponds 
with the net proceeds to be divided. 


Lrap’s, A $3090 ASSETS, $8650 A $3000 x .4@ = $1200 
B 6250 Expenses, 650 B 6250 x .46 — 2500 
C $80 ise C 800 x 40—- 320 
B 9950 Net-pis:, anee D 9950 x .40— 3980 


Scae $000 — 20000 — .40 


Total, $20000 Rate, 40 4, Proor, $8000. 


408. Ruru.—1. Find the rate of dividend, by divid- 
ing the net preceeds of the assets by the total of liabilities. 

2. Find each crediter’s dividend, by multiplying his 
claim by this rate. 

406. What is a Bankrupt? What is meant by the Assets of a bankrupt? By 


his Liabilities ?—407. When a person becomes bankrupt, what is usually done? Go 
threugh the example, explaining the several steps and proof.—408, Recite the rule. 


a a eee AS Paes ares See ve 
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oT BANKRUPTCY. 

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EXAMPLES FOR’ PRAOTIOER. 

1. A merchant becomes insolvent, owing A $375.50, B $1106,. 

C $4168.75, D $3725, and E $8630.75. His assets realize $11400, 
and the assignee’s charge is $600. What is the rate of dividend, 
and what each creditor’s share ? Ans. Rate, 60%. 
2. Harrison & Co. having failed, their liabilities are found to 
be $71600. Their assets consist of goods that sell for $9815; 
debts collectible, $17005; house and lot, worth $7250. The as- 
signee’s charge is 5% on the assets, and other expenses amount to 
$146.50. What % can they pay, and how much will Ira Jones 
receive, to whom they owe $12500 ? Last ans. $5625. - 
8. S becomes insolvent, owing $62000, and having $14200. 
assets; the expenses of settling are $560. How much can he 
pay on a dollar? What is P’s dividend on a claim of $1400? Q 
receives $275 ; what was his claim ? Last ans. $1250. 
4, A bankrupt settled with his creditors for 35c. on a dollar. - 
B received a dividend of $5075, and 0 54% of that amount; what 
were their respective claims ? Ans. O's, $725. 
5. The assets of a bankrupt are $42000. He owes V $17000, W 
$24150, X $37140.75, Y $28000.50, and Z $10708.75. Y becomes 
assignee, and receives 4% on the assets for his services; the other 
expenses of settling are $1320. What is each creditor’s share— 
Y’s to include his percentage as assignee? Ans. Y’s, $11018.50. 


(ee es 


CTWLA PTE Rak AW 
INSURANCE. 


409, Insurance is a contract by which, in considera- 
tion of a certain sum paid, one party agrees to secure 
another against loss or risk. 

410, There are different kinds of Insurance :— 

_ Fire Insurance secures against loss or damage by fire; 
Marine Insurance, against the dangers of navigation; 


INSURANCE. 257 


Accident Insurance, against casualties to travellers and 
others. Health Insurance secures a weekly allowance 
during sickness. Life Insurance secures a certain sum, 
on the death of the insured, to some party named in the 
contract. 

411. The Underwriter is the insurer,—the person or 
company that takes the risk. 

The Policy is the written contract. 

The Premium is the sum paid the underwriter for 
taking the risk. In the case of Fire and Marine Insur- 
ance, it is reckoned at a certain % on the sum insured. 


‘412. The rate is sometimes given at so many cents on $100, in 
stead of on $1. In that case, be careful to write the decimal properly. 
20 cents on $1 is written .2; on $100, .002. 45c. on $1 is .45; on $100, 
0045. 

Insurance is usually effected with companies. Some companies, to 
guard against fraud, will not insure to the full value of the property. 
Different rates are charged, according to the risk. In case of loss, the 
underwriters may either replace the property or pay its value. Only, the 
‘amount of actual loss can be recovered. 


413. The principles of Percentage apply to Insurance 
(Fire and Marine). The sum insured is the base; the 
premium is the percentage, reckoned at a certain rate. 
Hence, according to § 321, the following 

Rures.—lL. Zo find the premium, multiply the sum tn- 
sured by the rate. 

I. To find the rate, divide the premium by the sum in- 
sured. | 

Ill. Zo find the sum insured, divide the premium by 
the rate. 

EXAMPLES FOR PRAOTIOE. 


1. Insured a house for $10000, and furniture for $5000, at the 
rate of 30c. on $100; $1 being paid for the policy and survey, 
what does the insurance cost ? Ans. $46. 


409. What is Insurance ?—410, Name the different kinds of Insurance, and state 
against what each secures the insured._411. What is meant by an Underwriter? 
What is the Policy? What is the Premium ?—412, What caution is given as te 
writing the rate? How do some companies try to guard against fraud? In case 
ef loss, what may the underwriters do ?—413, Reeite the rules. 


258 EXAMPLES FOR PRACTICE. : 


2. At 4 of 1%, what is the premium on $8000? On $7250? 

At $%, what is the premium on $2200 ? First ans. $40. 

3. A factory and its contents, worth $72000, are insured for 

$ of their value, at 384 per cent. The whole is consumed. How 

much will the owner receive, and what will be the actual loss to 

the underwriters ? ‘3 Last ans. wiiiaie 
The actual loss is the sum they have to pay, less the premium. 

4, A merchant insures 1200 bar. of flour, worth $8 a barrel, 
for their full value, at $%. A fire occurring, only 450 barrels are 
saved. What premium does the merchant pay, how much will 
he receive from the company, and what will be their actual 
loss 2 Second ans. $6000. 

5, A vessel valued at $90000, and its cargo worth $55000, are 
insured for half their value, at 24%. What is the premium, in- 

/ eluding $1 for policy ? 

6. Insured $9000 worth of goods for } of their value, at 84% 
They were damaged by fire to the extent of $1250. What was 
the premium, how much did the underwriters pay the insurer, and 


what was their actual loss ? Last ans. $1212.50. 
7. The premium on a house, at + of 1%, cost me $20; what was 
»f the sum insured? (See Rule IIIL., § 413.) Ans. $6000. 


cee, 8. Paid for insuring a hotel for 3 of its value, $151. The rate 
being 75c. on $100, and the policy costing $1, what was the hotel 
worth ? Ans. $30000. 


As the policy cost $1, the premium was $151—$1, or $150. T5c. on $100 = .0078, 
rate. Apply Rule III., to find the sum insured, and this will be 2 of the value of 


the hotel. 
9. Paid $18 for insuring $9000; what was the rate? (See 
Rule IL, § 418.) Ans. 4 of 1 per cent. 


10. Paid $400 for insuring a factory, worth $48000, for 2 of its 
value; what was the rate ? 

11. An underwriter agrees to insure a hotel, worth anes 
for a sufficient sum to cover its value and the premium. The 
rate being 1%, for how much must he insure it ? 


Analogous to Example 2, § 323. As the rate is 1 per cent. of the sum to be in- 
sured, the value of the hotel, $24000, must be 99 per cent. of this sum. Then by 
ge Rule IIL, § 321, $24000 + .99 = $24242.42 Ans. 


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ACCIDENT INSURANCE. '- 959 


12. For how much must a schooner be insured, to cover its 
value, $15000, and the premium, the rate being 14%? What will 
the premium amount to ? Last ans. $228.43. 

18. Paid for insuring the full value of a ship and cargo, at 1%, 
$450. If the cargo was worth half as much as the ship, what 
was the value of the ship ? of 


414, Accipent Insurancre.—Insurance against acci- 
dents is effected by paying (in advance) an annual pre- 
mium, in consideration of which the underwriters give 
the insured a certain allowance per week in case he is 
disabled by an accident, or pay his heirs a specified sum 
if he is killed. 

14, A party paying $12 premium annually, in the third year 
for which he insures, is disabled by an accident for 13 weeks, 
during which time he receives $10 a week. How much more 
does he receive than he paid for premiums ? Ans. $94. 

15. A person who has paid five annual premiums of $30 each, 
is killed by an accident. His family receive $5000. Not reckon- 
ing interest, what is the loss to the underwriters ? 

16. A railroad conductor insures for $60 a year, his weekly 
compensation in case of a disabling accident to be $50. In the 
tenth year, he is laid up by an accident for 4 wecks; does he gain 
or lose by insuring, and how much, leaving interest out of ac- 
count ? Ans. Loses $400. 


415.—Lirre Insvrancu.—Life Insurance is effected by 
paying (in advance) an annual premium during life or for 
a term of years, in consideration of which the underwriters, 
on the death of the insured, pay a certain sum to his heirs 
or some party named in the policy. 

416. The rates of life insufince depend on the age at 
which one begins to insure, apd are fixed at a certain sum 
on every $100 or $1000 insuted. They differ but little in 
different companies, being based on the Expectation of 


414. How is Accident Insurance effected ?—415. Howis Life Insurance effected ? 
~416, On what do its rates depend? How are they fixed? On what are they based ? 


260 LIFE INSURANCE. 


Life,—that is, the average number of years that persons 
at different ages live, as shown by statistics. 


417, Rutx.—To find the premium in life insurance, 
multiply the premium on $100 or $1000 by the number 
of hundred or thousand dollars insured. 


17. What annual premium must person, aged 80 when he 
begins to insure, pay for a life policy of $5000, the rate being 
$2.3023 on $100 ? Ans. $115.12. 

18. At the age of 40, a gentleman insures his life for $3000, 
payment of premiums to cease in ten years. The rate is $57.959 
on $1000. If he dies at 55, how much more will his family re- 
ceive than he paid for premiums ? Ans, $1261.20. 

19. On his 40th birth-day,'a clergyman insures his life for 
$6000, payment of premiums to cease when he is 65. ‘The rate is 
$35.12 on $1000. If he dies aged 45 years 1 month, how much 
more than the premiums paid will his heirs receive? 

Ans. $4735.68. 

20. A farmer insured his life for $1750, at the rate of $3.66 
on $100. Just 9 months afterwards he died. Taking interest on 
the premium (at 6%) into account, how much was gained by in- 


suring ? Ans. $1683.07. 
AY ctateBeee ys oe tiald aA 
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CHAPTER XKVID  o¥ 
TUX Bhat sae | 


418. A Tax is a sum assessed on the person, property, 
or income of an individual, for the support of government. 

When assessed on the person, it is called a Poll-tax, 
and is a uniform sum on each male citizen, except such as 
may be exempted by law. 

When assessed on the property, it is called a Property- 
tax, and is reckoned at a certain rate on the estimated 
value. 


417, Recite the rule for finding the premium in life insurance,418, What is a 
Tax? Name and define the three kinds of taxes, 


TAXES. 261 


When assessed on the income, it is called an Income- ,/ 


tax, and is computed at a certain %. 

419, Taxable property is either Real or Personal. 

Real Estate is fixed property; as, lands, houses. 

Personal Property is that which is movable; as, cash, 
notes, ships, furniture, cattle, &c. 

420. An Assessor is an officer appointed to estimate 
the value of property and tax it in proportion. 

421, AssESSMENT oF TaxxEs.—In assessing a property- 
tax, an Inventory, or list, of all the taxable property, real 
and personal, with its estimated value, must first be made 
out.. If there is, besides, a poll-tax, a list of polls (that is, 
of persons liable to said tax) must also be drawn up. The 
poll-tax having been fixed, the rate of property-tax must 
then be found, and lastly each man’s tax. 

Ex. 1.—A tax of $6402 is to be raised in a certain town, 
containing 480 polls, which are assessed $1 each. The real 
estate of said town is valued at $878500, the personal prop- 
erty at $108500. What will be the rate on $1,—and what 
will be A’s tax, who pays for 4 polls, and whose real estate 
is inventoried at $5500, his personal property at $1250? 


$878500 + $108500 = $987000, total taxable property. 
$l x 480 = $480, total poll-tax. 
$6402 — $480 = $5922, property-tax to be assessed. 
By Rule IL, § 321, $5929, + 987000 = .006, rate. 

$5500 + $1950 — = $6750, A’s taxable property. 

$6750 x .006 = $40.50, A’s property-tax. 

$1 x4=— $4, A’s poll-tax. 
$40.50 + $4 = $44.50, etal A’s tax. 


9 Rutz.—1l. Zo find the rate of property-tax, divide 
the sum to be raised, less the amount assessed on polls, by 
the value of the taxable property, real and personal. 


v7 


419. How many kinds of taxable property are there? What is Real Estate? 
What is Personal Property ?—420. What is the business of an Assessor ?—421. In 
assessing a property-tax, what must first be made out? If there is, besides, a poll- 
tax, what must be done? What are the next steps? Go through the given ex- 
ample, explaining the steps.—422, Recite the rule. If there is no poll-tax, what 
must be done ? 


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262 TAXES, 


2, To Jind each man’s tax, multiply his taxable prop- 
erty by the rate, and to the product add his poll-tax. 

If there is no poll-tax, the whole amount to be raised must be divided 
by the value of the taxable property. 

493, If the given amount to be raised does not include the expense 
of collecting, the whole sum needed, including this expense, must first be 
found, by dividing the given amount by $1 diminished by the rate % to be 
paid for collecting. 

Thus, in Example 1, let the expense of collecting, 247%, not be in- 
cluded in the $6402 named; then, as $1 raised would net but $.975, 
there would have to be raised as many times $1 as $.975 is contained 
times in $6402. In other words, we should have to divide $6402 by $1 
diminished by .025, the rate paid for collecting. 

424, After finding the rate as above, assessors usually 
construct a Table, from which, by adding the amounts 
standing opposite to the thousands, hundreds, tens, and 
units of any given sum, they can readily determine the 
tax it must bear—more readily, as a general thing, than 


by multiplying by the rate. 
Assessor's Table for a rate of .006. 
g.006 | $10 $.06 | $100 $0.60 | $1000 g 6. 


$1 
2 .012 20 12 200 1.20 2000 12, 
3 .018 30 eke 300 1.80 8000 18. 
4 024 40 24 400 2.40 4.000 24. 
5 030) 50 80 500 3.00 5000 80. 
6 .036 60 06 600 3.60 6000 36. 
7 .042 70 42 700 4,20 7000 42. 
8 .048 80 48 800 4,80 8000 48. 
9 054 90 54 900 5.40 9000 54. 


2. Find by the Table what tax B must pay on $7560, 
Opposite $7000 we find $42.00 


500 “  — 8.00 
(9 60 (9 (<9 0.86 
Total for $7560, $45.86 Ans. 


3, What is O’s tax on $425, and 3 polls, at $1 each? 
D’s, on $900 real estate, $650 personal property ? 

K’s, on $2820 real estate, $710 personal, 1 poll? 

423. If the given amount to be raised does not include the expense of collecting, 


how must we proceed ? Dlustrate this in the case of Ex. 1.—424. After finding the 
~ rate as above, what do assessors usually construct? Show how the Table is used. 


TAXES, 263 


4, The people of a certain town have to raise a tax of $4656, 
besides the expense of collecting, which is 3% (see $423). The 
inventory shows real estate valued at $401250, and personal prop- 
erty at $98750. There are 400 polls, assessed at 75c. each. 

Find the rate on $1, draw out a Table like that on p. 262, and 
from it determine the tax of the following parties :— 

G, who pays on $3460 and 2 polls, Ans. $32.64. 

H, on $1975 and 4 polls. 

I, on $2000 real, $800 personal, and 8 polls. 

J, on $1750 real, $640 personal, and 1 poll, 


5. A tax of $7493.25:'is to be raised in a certain village, 
in addition to the expense of collecting, which is 3%. The real 
estate is valued at $497050, the personal property at $120950. 
There is no poll tax. What is the rate? Ans. .0125. 

How much must O pay, on a house and lot valued at $3500, 
furniture appraised at 4+ of that amount, and a pair of horses in- 


ventoried at $250? Ans. $53.125. 
What is P’s tax on $1500 worth of personal property, and an 
equal amount of real estate? Ans. $37.50. 


At what is Q’s property in the village valued, if his tax is $35? 
Ans. $2800. 
At how much are R’s reak estate and. personal property re- 
spectively valued, if the value of the former is three times that 
of the latter, and R’s tax is $45? 
Ans. Real estate, $2700; personal, $900. 
6. It is voted in a certain town to raise by tax $9250, ex- 
clusive of the collector’s commission, which is 5%. The town 
contains personal property valued at $150220, and real estate 
valued at $639600. Poll tax, 50c. on 518 polls. 
What is the whole amount to be raised ? Ans. $9736.84. 
How much is the callepter’ s commission on the whole? 
What must W pay, on $5225 real estate, $975 personal, and 


5 polls? Ans. $76.90. 
If X’s tax is $40.20, and he pays on 6 polls, what is his prop- 
erty valued at? Ans. $3100. 


264 DUTIES. 


CHAPTER XXVIII. 
DUTIES. 


426. Duties, or Customs, are taxes on goods imported 
from foreign countries, levied for the support of the Na- 
tional Government. 

427. A Custom-house is an office established by goy- 
ernment for the collection of duties. A port containing 
a custom-house is called a Port of Entry. 

428. Duties are either Specific or Ad valorem. 

A Specific Duty is a fixed sum imposed on each ton, 
pound, yard, gallon, &c., of an imported article, without 
regard to its cost. 

An Ad valorem Duty is a percentage on the cost of an 
imported article in the country from which it was brought. 
Ad valorem means on the value. 

429, An Invoice is a statement in detail of goods ship- 
ped, their measure or weight, and cost in the currency of 
the country from which they were brought. 

430. Before computing duties, certain Allowances, or 
Deductions, are made :— 

Tare is an allowance for the weight of the box, cask, 
&c., containing the goods; Leakage, for waste of liquids 
imported in casks; Breakage, for loss of liquids imported 
in bottles. 


Tare is estimated either at the rate specified in the invoice accompa- 
nying the goods, or according to rates adopted by Act of Congress, 
differing for different articles. 

For Leakage 2% is allowed; for Breakage, 10% on beer, ale, and 
porter, in bottles ; be ¢% on other liquids, we i sa + donee » bottles being 
estimated to contain oR gallons. 


426. What are Duties, or Customs ?—427. What is a Custom-house? What isa 
Port of Entry ?—428, Name the two kinds of duties. What is a Specific Duty? 
What is an Ad valorem Duty ?—429. What is an Invoice ?—430, Name and define 
the allowances made before computing specific duties. How is Tare estimated ? 
How much is allowed for Leakage? How much for Breakage? 


DUTIES, 265 


In stead of computing by these fixed rates, the weight of the box, 
&c., and the amount lost by leakage and breakage, are sometimes ascer- 
tained by actual trial and allowed for accordingly. 

In these allowances, reject a fraction less than 4; reckon 4 or more 
as 1—In custom-house computations, allow 112 lb. to a ewt. 

431, Gross Weight is the weight of goods, together 
with that of the box, cask, bag, &c., containing them. 

Net Weight is the weight of goods after allowances 
_have been deducted. 


432. Rutus.—l. Jo jind a specific duty, deduct allow- 
ances, and multiply the number of tons, pounds, yards, — 
gallons, &c., remaining, by the duty on one ton, pound, 
yard, gallon, de. 

Il. Zo find an ad valorem duty, multiply the tnvoice- 
value of the goods by the given rate. 


Duties are required to be paid in gold. 


EXAMPLES FOR PRAOTIOLE. 


1. What is the duty on a lot of silks, costing in our currency 
$14056, at 60%? When gold is at a prémium of 407, what sum 
in currency will pay said duty ? , Last ans. $11807.04.. 

2. Imported 150 casks of raisins, weighing 112 lb. each. The 
tare being 12%, and the duty 24 cents a pound, what is the duty 
on the whole in gold? When gold is at 180, what sum in cur- 
rency will pay it? Last ans. $480.48. 

3. Required the duty on 42 barrels of spirits of turpentine, 
containing 81 gallons each, leakage being allowed, and the rate 
being 80 cents per gal. Ans. $382.80. 

4, At 40% ad valorem, what is the duty on 846 lb. of sewing- 
silk, bought for $12 a pound? 

5. What is the duty on 6 casks of claret, holding 48 gal. each, 
invoiced at $1 a gal., allowing for leakage, the rate being 60c. a 
gallon ? Ans, $151.80. 


How are these allowances sometimes determined? How many pounds are 
allowed to 1 cwt., in custom-house computations ?—4381, What is Gross Weight? 
What is Net Weight ?— 432. Recite the rales. 

12 ; 


266 , DUTIES. 


6. The duty on cassia being 10 cents a pound, what must be 
paid on 175 packages of cassia, each weighing 35 lb., a tare of 97 
being allowed ? 

7. What is the duty on 12 cases of brandy, containing 1 dozen 
bottles each, the usual allowance being made for breakage, and 
the rate being $2.00 a gal. ? Ans. $62. 

8. At 12 cents per 100 Ib., what is the duty on 89 bags of salt, 
averaging 100 lb. gross weight, tare 27? | 

9. A merchant imported 10 hhd. of sugar averaging 1185 Ib., 
and 8 hhd. of molasses holding 63 gal. each. <A tare of 124% is 
allowed on the sugar, and leakage on the molasses. What is the 
duty on the whole, the rate on the sugar being 2c. a lb., and on 
the molasses 5c. a gal. ? Ans. $309.85, 


an 
bes 


CHAPTER XXVIII. 
EQUATION OF PAYMENTS. 


433. Equation of Payments is the process of finding 
when two or more sums due at different times may be 
paid at once, without loss to debtor or creditor. ‘The 
time for such payment is called the Equated Time. 


Ex. 1.—A owes B $1000, of which $100 is due in 2 
months, $250 in 4 mo., $350 in 6 mo., and $300 in 9 mo. 
If A pays the whole sum at one time, how long a credit 
should he have ? 


The use of $100 for 2 mo. 
Pee i Se tae) TOR ae 111: 
ie ee SOD) Or 0 10. 
eS $300 for 9mo. 


use of $1 for 100 x 2, or 200 mo. 
“ $1 “ 250 x 4, or 1000 mo. 
“ $1 “ 350 x 6, or 2100 mo. 
“ $1 “ 3800 x 9, or 2700 mo, 


r 


a 
nn 


HU Tt 


n~ 
n~ 


Hence A is entitled to the use of 1000 ) 6000 mo. 
$1 for 6000 mo., or $1000 for ‘Ans. 6 mo. 
yuo of that time, or 6 mo. : 


433, What is Equation of Payments? What is meant by the Bquated Time? 
Go through Ex. 1. 


EQUATION OF PAYMENTS. ee 267 | 


434, Rute.— To equate two or more payments, multi- 
ply each payment by its time, and divide the sum of the 
products by the sum of the payments. 


The times of the several payments must be in the same denomina. 
tion, and this will be the denomination of the answer. 
Less than 4 day in the answer is rejected ; 4+ day or more counts as 1, 


435, If the date is required, reckon the equated time forward from 
the given date, 


iix, 2.—July 9, 1866, C becomes indebted to D for a 
certain sum; 4 is to be paid in 6 months, 1 in 8 mo., and 
the rest in 12 mo. At what date may he equitably pay 
the whole ? 


aX, 6. = 2 

Use the fractions representing the amounts 4x 8=2 

as in Ex. 1. The equated time being 9 months, xs X 12 = 5 

payment should be made 9 months from July 9, ers) 9 

1866,—that is, April 9, 1867. ORS ae eas 
oe Ans. 9 mo. 


EXAMPLES FOR PRAOTICE. 


| Pe 
1. A merchant has the following sums due from a customer; 
$300 in 2mo., $800 in 5mo., and $400 in 10mo. Find the 
equated time. Ans. 5mo. 22 da, 

2. E owes F $1200, $200 of it payable in 2 mo., $400 in 5 mo., 

and a rest in 8mo. What is the equated time ? 

. A trader bought goods, Aug. 1, 1866, to the amount of 
ars : for $ of the bill he was to pay cash; 4 of it he bought on 
6 months’ credit, and the rest on 10 months. On what,day may 


he equitably pay the whole ? Ans. Feb. 6, 1867. 


The cash payment must be added with the others, but its t +x0=0 
product is 0. avy 


4. One person owes another a certain sum, + of which is due 
in 34 mo., + in 44. mo., 4 in 5 mo., and the balance in 8mo. What 
is the equated time ? Ans. 5mo. 7 da. 

5. Jan. 1st, I owe a friend $100 cash; $150, payable Feb. 5; 
and $300, payable April10. It being leap year, on what day 
may I fairly pay the whole at once ? Ans. Mar. 5. 

The Table on p. 156 will assist in finding the number of days. 


434, Recite the rule.—435, If the date is required, what must be done ? 


268 EQUATION OF PAYMENTS. 


. 6..Lquate the folowing payments: $400 due in 15 days, $600 
in 20 days; $1000 in 60 days; $350 in 90 days. 

7. A farmer, on the 1st of March, bought some land for $1000. 
He agreed to pay $250 cash; $250 on the 8d of the following May; 
$250, July 4; and $250, Sept. 15... He prefers paying the whole 
at once ; when should it be? Ans. June 6. 


Ex. 8.—Suppose $700 to be due in 6mo. At the ex- 
piration of 8 mo., $100 is paid on account; and at the 
end of 5 mo., $300.. How long after the six months ex- 
pire should the balance be allowed to stand, in considera- 
tion of these prepayments ? 


On the principle applied in Ex. 1, the 100 x 3 = 300 
creditor gets the use of what is equiva- 800 x 1 = 300 
lent to $1 for 600 mo.; the debtor is, there- ie ae 
fore, also entitled to the use of $1 for 400 600 
600 mo., or $300 (the balance) for zy of _ 700 — 400 = 800 
600 mo., or 2 mo. 600 + 800 = 2mo. Ans. 


436, Rute.— When partial payments have been made 
on a debt. before it is due, to find how long the balance 
should. remain. unpaid, multiply each payment by the 
time it was made before falling due, and divide the sum 
of these products by the balance. 


9. A person. owes $1000, due in 12 mo. At the end of 3 mo. 
he pays $100, and one month afterwards $100. How long be- 
yond the 12 mo. should the balance stand ? Ans, 2mo. 4 da. 

10. $1496.41 is due in 90 days. 84 days before it falls due, 
$500 is paid, and 52 days after the first payment $502.50. How 
long after the 90 days, before the balance of the debt should be 
paid ? Ans. 118 days. 

_ 11. A lent B $200 for 8 months, and on another occasion 
$300 for 6 months. How long should B lend A $800, to balance 
these favors? Ans. 44mo. 

12. A credit of 6 mo. on $500, one of 4 mo. on $1000, and one 
of 8 mo. on $400, are equivalent to a credit on how many dollars 
for 12 mo. ? Ans. $850. 


Analyze Iix. 8.—136. Recite the rule for finding how long a balance should 
stand, when partial payments haye been made on a debt before it is due, 


EQUATION OF PAYMENTS. 269 


13. T. Hoe buys goods of G. A. Rand, as follows :— 
1, May 1, bill of $600, on 3 mo. credit. 
Jee Maye 15,“ -“ BBO0. . wlanoe si“ 
Bettie 1. SP “S50G Orton 
4, June 9, “ “* $900, for cash. 
Rand agrees to take Hoe’s note for the whole, for 30 
days, with interest. When should the note be dated ? 


Here the terms of credit begin at different dates. We must first find 
when each bill falls due, by reckoning forward from its date the term of 
credit. 


Term of credit. Diie. Payt. Days. Product. 
1. 8 mo. from May 1, Aug. I, $600 x 538 = 31800 
2. 4mo. from May 15, Sept. 15, $800 x 98 = 78400 
3. 6mo. from June 1, Dec. 1, $500 x 175 = 87500 
4. Cash payment, June 9, $900 x1, O= 0 
$2800 197700 


197700 -+ 2800 = 70}. 
Equated Time, 71 days. 

Since there is no uniform date to reckon from, as in the former ex-. 
amples, we take the earliest date on which a payment falls due, June 9, 
and find the number of days from that time to the date when each pay- 
ment falls due, writing it opposite the payment it belongs to, as in the 
4th column above, Then finding the products and dividing as. before, 
we get 71 days for the equated ‘time, which must be reckoned forward 
from the standard date, June 9. 

21 days remaining in June, 
a Xt in July. 
71 — 52 = 19. . Ans. August 19. 

We might have assumed the latest date at which a payment fell due, 
Dec. 1, as a standard, proceeded as above, and reckoned the equated time 
so found back from that’ date. The result would have been the same. - 
The operation may always thus be proved. 


437. Rute.—TZo equate payments when the terms of 
credit begin at different times, jind the dates when the 
several payments become due. rom the earliest of these 
dates, as a standard, reckon the number of days to each 
of the others. Then find the equated time as before, § 434, 
and reckon tt forward from the standard date. 

Explain Ex. 13, How does this differ from the preceding examples? Why do 
we assume the earliest date as a standard? What other date might have been as- 


sumed? How may the operation be proved ?—437. Recite the rule for equating pay- 
saents, when the terms of credit begin at different times, 


H 


270 EQUATION OF PAYMENTS. 


_ We may shorten the multiplication, without, materially affecting the 
result, by rejecting less than 50 cents in any payment, and calling 50 
* cents or over, $1. 

14, Bought goods of Parsons & Co., on different terms of 
credit, to the following amounts: March 6, $275.50, on 80 days; 
March 31,. $560, on 3 months; April 10, $820.10, on 60 days; 
May 3, $515, on 4months; May 9, $1225.40, on 6months. At 
what date may the whole be discharged at once? Ans. Aug. 14. 

15. Harvey Bolton is indebted to a silk-house for goods bought, 
as follows:—June 1, $842, on 6 months; June 2, $1500, on 4 
months; June 8, $1875.75, on 3 months; June 4, $400, on 6 
months; June 5, $750, cash. In stead of paying the items sepa- 
rately when due, Bolton gives his note, without interest, for the 
whole; for how much should his note be drawn, and when should 
it mature? Last ans. Sept. 19. 

16. Cranston & Miner have sold goods to Henry 8. Owens, as 
follows :—Novy. 1, 1865, on 6 months, $1200; Nov. 5, on 4 mo., 
$800; Nov. 30, on 8 mo., $440.96; Dec. 3, on 90 days, $650; 
Dec. 10, on 2mo., $1120.25; Dec. 24, on 6mo., $347. Owens 
proposes to discharge the whole at one payment; when should it 
be made ? Ans. March 22, 1866. 

17. Sold a customer the following goods: Aug. 2, 2 dozen 
overcoats, @ $25 each, on 60 days’ credit; Aug. 4, 6 dozen boys’ 
sacks, @ $8.50, and 12 dozen boys’ pants, @ $5, on 90 days; Aug. 
5, 4dozen cassimere pants, @ $12, on 90 days; Aug. 6, 6 dozen 
vests, @ $3.25, on 4months. When should a note for the whole 
amount, without interest, mature ? Ans. Oct: 29. 

Averaging Accounts. ‘ 

438, An Account is a statement of mercantile transac- 
tions, its left side (marked Dr.) being appropriated to 
debits, and its right side (marked Cr.) to credits. The 
difference between the sum of the debits and that of the 
credits is the Balance of the Account. 


338. What isan Account? What is meant by the Balance of an Account? 


f 
AVERAGING ACCOUNTS. DAE i 


439. Averaging an Account is the process of finding 
the equitable time for the payment of the balance. 


Those accounts only need averaging, in which items occur bearing 


ronk) 


interest from their date, or from the expiration of their terms of credit. 


440, Finding the Cash Balance of an account is find- 
ing what sum will balance the account at any given time, 
interest being allowed on the several items. 

Ex. 1.—Average the following account, supposed to 
be taken from the Ledger of Stephen Stewart :— 


Dr. Mosrs T. Marsu. Cr. 
1866 1866 : 
May 8 To Merchandise | $900 || Apr. 3 | By Merchandise | $200 
be al Ae i: 850 eed re & Rint Ss 400 
(<9 15 ce «¢ 610 133 17 cc ce 500 
June 1} “ § 400 || May 15 |. “ Cash 450 


Marsh owes Stewart $1210, as is found by balancing the account. _ 
When is it equitably due? Or, if Stewart Tova his note for the balance, . 
when should it be dated ? LQ 2s 

Take the earliest date on cither side of the account, April 3, as the 
standard. Then, according to the principle already explained, the inter- 
est on all the debits from this standard date to the times they severally 
fall due would equal the interest of $1 for 113870 days (see operation 
below); that on the credits would equal the interest of $1 for 28700 
days. There is, therefore, an excess of interest in favor of the debits, 
equal to the interest of $1 for 85170 days—or of $1210 (the balance of 
account, on the debit side) for yao of 85170 days, or 70 days. Hence 
Marsh is entitled to retain the balance he owes, till the expiration of 70 
days from the standard date, April 8,—or June 12. 


Debits, 900 x 85 = 31500 Credits, 200: %°°0 = 0 


850 x 89 = 33150 400° X'S Fi=S:-2500 
610 x 42 = 25620 500 x 14.= 7000 
400 x 59 = 28600 450 x 42 = 18900 
2'760 113870 1550 28700 
1550 28700 
Balance, 1210) 85170 Excess of debit products. 
Averaged time, 70 days, Date, June 12. Ans. 


Had the excess of interest and the balance of account stood on oppo- 
site sides, we should have had to count the 70 days back from the stand- 
ard date. . 


439. What is Averaging an Account? What accounts need averaging ?—440, 
What is meant by finding the Cash Balance of an account? Explain Ex,1. Under 
what circumstances would we have had to count the 10 days back? 


272 FINDING THE CASH BALANCE. 


441, Ifa credit were allowed on each of the merchandise items, we 
should have found when each item became due, and used those dates in 
stead of the dates of the transactions. Thus :— 


Ex. 2.— Average the account presented in Ex. 1, allow- 
ing each merchandise item a credit of 3 months. 
Find when each item falls due. May 15 is the standard date. 


DEBITS. CREDITS. 


Due Aug. 8,$900 x 85= 76500 Due July 8, $200 x 49= 98090 
1 Aug, 12, 850 x 89= 75650  “ July10, 400 x 56= 22400 
“ Aug. 15, 610x 92= 56120 “ July 1%, 500 x 63= 81500 


“ Sept. 1, 400 x 109= 438600 “May 15, 450 x O= 0 
B60 ser 251870 1550 63700 
1550 63700 
Balance, 1210) 188170 Excess of debit products. 
Averaged time, 156 days. Date, Oct. 18. Ans. 


442, Casu Batancr.—What is the eash balance of 
the account presented in Ex. 1, due Aug. 20, allowing 3 
months’ credit on each merchandise item, and interest 
at 6%? 

We have just found, in Ex. 2, that the balance of $1210 is due Oct. 
18. The cash balance on August 20th is therefore the present worth of 
$1210, due Oct. 18,—that is in 1 mo. 28 days. This, by § 378, is found 
to be $1198.42. Ans. 

Had the given date of settlement fallen after the averaged date, Oct. 
18, we should have added interest to $1210 for the interval. 

With Interest Tables, which accountants wniversally use, the second 
method given in the rule below will be found the more convenient. 

From the above examples we derive the following rules :— 

443. Rutzes.—L. To average an account, take the ear- 
liest date on either side as a standard, and multiply each 
item by the number of days between the time when it falls 
due and the standard date. Divide the difference between 
the sum of the debit and that of the credit products by the 
balance of the account. The quotient will be the averaged 
time. Steckon this forward from the standard date, if 
the excess of products is on the same side with the balance 
of account ; if not, backward. 

441. What must we do when a credit is allowed on the merchandise items? 
Explain Ex. 2.—442. How may we find the cash balanée of the account presented in 


Ex. 1, due Aug. 20?—443, Recite the rule for averaging an account. For finding the 
cash balance. 


FINDING THE CASH BALANCE, 273 


Il. Zo find the cash balance, average the account, and, 
if the given date of settlement falls before the averaged 
time, find the present worth of the balance of account for 
the interval ; if after, add interest for the interval. 

Or, find the interest on each item from the date it falls 
due to the time of settlement ; write it on the same side 
of the accouni as its item, if the item falls due before the 
date of settlement,—if not, on the opposite side. Find the 
balance of interest, and add it to the balance of the ac- 
count if the two balances stand on the same side ; if not, 
subtract tt. 


3. Average, and find cash balance Mar. 1, 1866, at 74%. 


Dr. REvBEN THOMPSON. Cr. 
1866 1866 
Jan. 2 | To Cash $1200 || Jan..10 | By Merch., 4 mo, | $1000 


Feb. 5 | ‘‘ Merch. 60da.} 1400 |) “ 18 | “ Merch., 6mo. | 1160 
“8 | “ Merch., 3 mo. | 1500 || Feb. 2 | “ Merch., 3mo. | 1250- 
“ 10 | “ Merch., 60da.| 2000 “6 | “ Merch., 6 mo, | 1300 


Ang, § Balance of acct., $1890, due June 23, 1865. 
* ( Cash balance, March 1, 1866, $1457.03. 


4. Average, and find cash balance Jan. 1, 1866, at 6%: 
Dr. Arperr B. Conner. Cr. 


1865 | Ly |babeas 
Sept. 3 | To Merch., 3 mo. | $750 || Oct. 1 | By Merch., 4:mo. | $200 
“ 20 | * Merch.,4mo, |; 610 = 20-.)°*** Cash 800 
Oct. 12 | “ Merch.,4mo, | 900 || Nov. 5 | “ Merch., 3mo. | 825 
Nov. 1 | “ Merch, 6mo.| 220 “ 12.| “ Merch., 6mo. | 540 
“10 |.“ Merch., 3mo,} 400 
aes Balance of acct., $1015, due Feb. 10, 1866. 
“*- 1 Cash balance, Jan. 1, 1866, $1008.45. 


MiscELLANEOUS Quzstions.—Recite the rules relating to Percentage, 
§ 321. Apply these rules to Interest, showing what corresponds to the 
base, and what to the percentage. Show how the rules of Percentage 
apply to Bank Discount. To Commission, To Bankruptcy, in ascer- 
taining the rate of dividend, and in finding each creditor’s share. To 
Insurance. To Assessment of Property Taxes, in determining the rate, 
and in finding each individual’s tax. To ad valorem Dutics, 


274 RATIO. 


CHAPTER XXIX. 
RAT LO. 


444. Ratio is the relation that one quantity bears to 
another of the same kind. It is represented by the quo- 
tient arising from dividing one by the other. The ratio 
of 8 to 2 is 4. 

445. Two quantities are necessary to form a ratio; 
these are called its Terms. 

The Antecedent is the first term of a ratio; the Conse- 
quent, the second. 

446, A ratio is either Direct or Inverse. It is Direct, 
when the antecedent is divided by the consequent ; In- 
verse, when the consequent is divided by the antecedent. 


When the word ratio is used alone, a direct ratio is meant. 


The direct ratio of 8 to 2is 4. The inverse ratio of 8 to2is}#. In 
either case, 8 is the antecedent, and 2 the popsar De. 


447, Ratio is expressed in two ways:—1l. By two dots, 
in the form of a colon, between the terms; as, 8:4. 2. In 
the form of a fraction; as, $. 

The two dots and the fractional line both come from the sign of di- 
vision —-. When the two dots are used, the line between is omitted ; 
when the fractional Jine is used, the two dots are omitted. 

8:4 is read the ratio of 8 to 4. 

448. A ratio being expressed by a fraction, of which 
the antecedent is the numerator and the consequent the 
denominator, it follows that the principles which apply 
to the terms of a fraction, S187, apply also to the terms 
of aratio, That is, 

_ Multiplying the antecedent multiplies the ratio, and 
dividing the antecedent divides the ratio. 


444, What is Ratio? By what is it represented ?—445. How many quantities 
are necessary to form a ratio? What are they called? Which is the Antecedent ? 
Which, the Consequent ?—446. What is the difference between Direct and Inverse 
Ratio? Give an example.—447. In how many ways is ratio expressed? Describe 
them. What is the origin of the two dots and the fractional line ?—448. State the 
three principles that apply to multiplying or dividing the terms of a ratio. 


Da 


RATIO. 275 


Multiplying the consequent divides the ratio, and di- 
viding the consequent multiplies the ratio. 

Multiplying or dividing both terms by the same nuim- 
ber does not alter the ratio. 

449, Fractions having a common denominator are to 
each other as their numerators. | 

vo: 3% = 7:9, or%. For, as we have just seen, dividing both terms 
of the second ratio, 7 and 9, by the same number, 10, does not alter their 
ratio.—The ratio between two fractions that have not a common denomi- 
nator, may be found by reducing them to others that have, and taking 
the ratio of their numerators. 

450. There is no ratio between quantities of different 
kinds; as, 8yd. and 41b. But a ratio subsists between 
quantities of the same kind, though of different denomi- 
nations. 


Thus, the ratio of 8 yd. (= 24 ft.) to 4ft. is 6. In such cases, to find 
the ratio, the terms must be brought to the same denomination. 


451. A Simple Ratio is one into which but two terms. 
enter. A Compound Ratio is the product of two or more 
simple ratios, the first term being the product of the an- 
tecedents, the second that of the consequents. 


Simple Ratios, 8: 4 The ratio compounded of these 
: three simple ratios is 
2:6 8x 9x 2:4 x 3 x 6, 


EXERCISE. 
1. Express the ratio of 27 to 9;. of 7 to 16; of 48 to 100. 
%. Read the following ratios :— 


144: 12 B24 6 lb. : 12 Ib. 4 owt. : 16 Ib. 

16 ; 288 be: $ 9 gr. : 4h er. 3 mi. : 20rd. 
tes 1000.7. 8 2mo. : 7.5 mo. 2 pt. : 16 gal. 
240: .8 $3.1 $5 : $.001 6 gt. : 50 bu. 


3. Find the value of the above ratios, when direct. 
4, Find the value of the above ratios, when inverse. 


449, What ratio do fractions having a common denominator sustain to each 
other? Prove this. Hence, how may the ratio between two fractions that have not 
a common denominator be found ?—450. How may we find the ratio between two 
quantities of the same kind, but different denominations ?—451, What is a cai 
Ratio? What isa Spe etre Ratio? Give an example. 


276 PROPORTION, 


CHAPTER XXX. 


PROPORTION. 


452. Proportion is an equality of ratios. 

The ratio of 8 to 4 is 2; the ratio of 6 to 3 is also 2. Hence the 
proportion, 8:4 = 6:3. 

453, Proportion is expressed in two ways:—l1. By the 
sign of equality. between the ratios. 2. By four dots, in 
the form of a double colon, between the ratios. 

8:4—=6:83) Read, 8 2s to 4 as 6 2s to 3. 
83463 t Or, the ratio of 8 to 4 equals the ratio of 6 to 3. 

454, Four quantities forming a proportion are calied 
Proportionals. The first two are called the First Coup- 
let; the last two, the Second Couplet. The first and 
fourth are called the Extremes; the second and third, 
the Means. 


In the proportion 8: 4:: 6: 3, 8 and 4 form the first couplet, 6 and 
3 the second. 8 and 8 are the extremes, 4 and 6 the means. 


455. Three quantities are in proportion when the Ist 
is to the 2d as the 2d to the 38d. 8:4::4: 2. 

A term so repeated is called a Mean Proportional be- 
tween the other two. 4 is a mean proportional between 
8 and 2, 

456. The product of the extremes, in every proportion, 
equals the product of the means. ‘Thus, in the last pro- 
portion, 8 x 2= 4x 4. Hence the following rules :— 

457. Rures.—I. To find an extreme, divide the product 
of the means by the given extreme. ; 

II. Zo find a mean, divide the product of the extremes 
by the given mean. 


452. What is Proportion ?—453, In how many ways is proportion expressed ? 
Describe them.—454. What are four quantities forming a proportion called? What 
are the first two called? The last two? Which, are the Extremes? Which, the 
Means ?—455. When are three quantities in proportion? What is meant by a Mean 
Proportional ?—456. What principle holds good in every proportion ?/—457, Give the 
rule for finding an extreme.. For finding a mean. 


ee ae ee 


SIMPLE PROPORTION. pag 


Ex. 1.—I'ind the 4th term of the proportion 8. 4::26:? 


Find the product of the means : 4 x 26 = 104, 
Divide by the given extreme: 104-8 = 13. Ans. 


Ex, 2.—F ind the 2d term of the proportion 8 : ?:: 26:13. 


Find the: product of the extremes: 8 x 13 = 104. 
Divide by the given mean : 104 + 26 = 4. Ans. 


EXAMPLES FOR PRAOTICE. 


Complete the following proportions :— 


1. 18: 54:: 206: ? Ans, OOU. | 14s-1o OF. bt... 2 O as 
2.°60.: 90 2374: 1.88 8. 2cwt. : 2ZO1b. +: $16 ; 2 
So. ES Ise 1233 9.99 Demy, Serb y ss Aas 
4, %7:80:24:1 10.6007: 72: 8°: 20 

5. 3pt.: 12pt.::2bu.:? Ans. Sbu. Tie Tra: 3 tts Sf code, 
6. 1qt.:?::1hr.:1da, Ans. 8pk. | 12. 450:80::1200:? _ 


Simple Proportion, or Rule of Three. 


458. A Simple Proportion expresses the equality of 
two simple ratios. Simple Proportions may be used te 
solve many questions in whith three proportionals are 
given and the fourth is required. 

As three terms are given, the rule for Simple Propor- 
tion is often called the Rule of Three. 


Kx. 1.—ff 8 yd. of cloth cost $40, what will 24 yd. cost? 


The terms of a couplet must be of the same kind. Hence, in forming 
a proportion from the above question, as the answer, or fourth term, is 
to be dollars, we take $40 for the third term. Then, since 24 yd. will cost 
more than 8 yd., we arrange the other two numbers go as to form an in- 
verse ratio greater than 1, by taking 24, the greater, for the second 
term, and 8, the less, for the first. The proportion then stands, 


Syd. : 24 yd. :: $40, the cost of 8 yd. : the cost of 24 yd. 
The 4th term is required ; we find it by Rule 1, § 457. 
24 x 40 = 960 960 — 8 = 120 Ans. $120. 


458,:What does a Simple Preportion express? To what questions do Simple 
Proportions apply? _ What is the rule often called? Explain Ex. 1.—459. How 
may cancellation-be brought to bear ?—460, Ivecite the rule. 


278 - SIMPLE PROPORTION, 


459. In solving questions in Proportion, equal factors, 
if there are any, in the Ist and 2d, or Ist and 38d terms, 
should be cancelled. Thus, in Ex. 1 :— 


Syd. : ZA yd. :: $40. 
3 


$40 x 8 = $120 Ans. 


460. Rutzt.—1l. Zake for the third term the number 
that is of the same kind as the answer. Of the two re: 
maining numbers, make the larger the second term, when 
Srom the nature of the question the answer should excced the 
third term ; when not, make the smaller the second term. 

2. Cancel equal factors in the first and second terms, 
or the first and third. Then multiply the means together, 
and divide their product by the given extreme. 


The first and second terms must be of the same denomination. If 
the third term is a compound number, it must be reduced to the lowest 
denomination it contains, and this will be the denomination of the answer: 


EXAMPLES FOR PRAOCTIOE. 


1. What cost 8 cords of wood, if 2 cords cost $9? Ans. $36. 

2. If 25 ib. of coffee cost $4.50, what cost 312 1b.? Ans. $56.16. 

3. Ifa railroad car goes 17 miles in 45 minutes, how far will it 
go in 5 hours at the same rate ? Ans. 1184 mi. 

4. Hlow long will it take $100 to produce $100 interest, if it 
produces $7 in one year? 7 | 

5. If 15 men can build a wall 12 ft. high in 1 wk., how many 
will be needed to raise it 20 ft. in the same time? How long 
would it take 5 men to raise it 20 ft. ? Last ans. 5 wk. 

6. What cost 9 hats, if 5 hats cost £4 5s.? Ans. £7 18s, 

7. If 7 tons of coal, of 2000 lb. each, last 834 months, of 30 days 
each, how. much will be consumed in 3 weeks? 

8. If 9 bu. 2pk. of wheat make 2 barrels of flour, how many 
bushels will be required to make 18 barrels ? 

9. If 5 bu. of potatoes last 8 adults and 2 children 40 days, 
how long, at the same rate, will they last 18 adults and 9 chil- 
dren, each adult consuming as much as 2 children? Ans. 16 days. 


EXAMPLES. 279 


10. How long will it take a steamboat to move its own length, 
if it goes 15 miles an hour and is 242 feet long? Ans, 11sec. 

11. How many times its own length will a steamboat move in 
eleven hours, if it is 242 ft. long and goes 15 miles an hour ? 

12. A reservoir has two pipes that can discharge respectively 
30 gal. and 15 gal. in one minute. How long will they be in dis- 
charging 15 hogsheads ? Ans. 21 min. 

13. If a man can mow 9 acres in 8} days, of 10 hours each, 
how many such days will it take him to mow 21 acres? 

14. An insolvent debtor owes $7560, and has only $3100 with 
which to make payment. How much should a creditor receive, 
whose claim is $378 ? Ans. $155. 

15. If =% of a ship is worth $2858, and #4 of the cargo is worth 
$6080, how much are both ship and cargo worth? 

16. How many yards of. oil-cloth, 14 yd. wide, will be needed 
to cover a certain floor, if 80 yd., # yd. wide, will cover it? 

17. If the earth moves through 860° in 865} days, how far 
will it move in a Junar month of 294 days ? Ans. 293% aT 

Compound Proportion, or Double vce 
Rule of Three. 


461. A Compound Proportion expresses the hee 
of a compound and a simple ratio. 

Compound Proportions are used in solving quasione 
that involve two or more simple proportions; hence this 
rule is often called the Double Rule of Three. Ae 

Ex.—If 6 men can mow 80 acres of grass in 5 days, / Z 
working 8 hours each day, how many acres can 4 men 
mow in 9 days, of 10 hours each ? 

As the answer is to be acres, we write 30 acres as the third term. We 
then take the other terms in pairs of the same kind—6 men and 4 men, 

5 days and 9 days, 8 hours and 10 hours, and form a ratio with each pair 


as if the answer depended on it alone, as in simple proportion. As 4 
men will mow less than 6 men, we take the smaller number for the sec- 


461. What does 2 Compound Proportion express? What‘is the rule for Com- 
pound Proportion often called? Why so? Explain the Example. 


Pes Ys Le ee A a ie 
Fe oe mane 
Ee et +See 


aS 
\. 


280 COMPOUND PROPORTION. “ 


ond term of the ratio, 6:4. As in 9 days they can mow more than in 
5 days, we-t#ke the greater for the second term, 5:9. As working 10 
-Noups*a day they can mow more than working 8 hours a day, we take 


the greater for the second term, 8:10. The proportion then stands, 


a 6 men : 4-men :7 30 acres Caiteelling, 6 | A 
a, _» days: 9 ays ee i §| 9 
Bee. 2 ey 50) in \ $ | Ap 5 
i g.-. Cabo _equal- secon Me ae Ma Z 3 
rn cect as i in Simple Proportion. 9 x 5 = 45 acres Ans? 


eZ 462. Rourz.—1. Take for the third term the number 
Pho is of the same kind as the answer. or the first and 
& second terms, form the remaining numbers, taken in pairs 
of the same kind, into ratios, making the larger nwmber 
the consequent when the answer, if it depended solely on 
the couplet in question, should ecneced the third term ; when 
not, make the smaller the consequent. 
“4 2. Cancel as in Simple Proportion. Multiply to- 
{gether the second and third terms that remain, and di-- 
vide their product by the product of the first terms. 


The first and second terms of each ratio must be brought to the same ~ 2 


denomination. If the third term is a compound number, it must be re- 
‘duced to the lowest denomination it contains, and this ill be the de- 
nomination of the answer. 


EXAMPLES FOR PRACTICE. 


; 1. If a person travels 800 miles in 17 days, journeying 6 hour: 
each day, how many miles will he travel in 15 days, journeying 1¢ 

hours a day? Ans, 4418; mi. 

at will be the weight of a slab of marble, 8 ft. long, 48 
ay. wide, and 5in., thick, if a slab of the same density 10 ft. long, 
8 ft. wide, and 8 in. thick, weighs 400 Ib. ? Ans. 7114 1b. 
v7 3. If the expenses of a family of 10 persons amount to $500 in 
23 weeks, how long will $600 support eight persons at the same 
rate ? Ans. 844 wk. 
4, 15 men, working 10 hr. a day, have taken 18 days to build 


462. Recite the rule for Compound Proportion, What reductions may have to 
be made ? 


EXAMPLES, 281 


450 yd. of stone fence. How many men, working 8 days, of 12 
hours each, will it take to build 480 yd. ? Ans, 30 men. 
5. If it takes 1200 yd. of cloth, £ wide, to clothe 500 men, how 
many yards, $ wide, will be needed for 960 men? - Ans, 82912 yd. 
6. How many pounds of wool will make 150 yd. of cloth, 1 yd. 
wide, if 12 ounces make 24 yd., 6 qr. wide? 
7. If the wages of 6 men, for 14 days, are $126, what will be 
the wages of 9 men, for 16 days? Ans. $216. 
8. If 100 men, in 40 days of 10 hours each, build a wall 380 ft. 
long, 8 ft. high, and 24 in. thick, how many men will it take to 
build a wall 40 ft.. long, 6ft. high, and 4 ft. thick, in 20 days, 


working 8 hours a day ? % ‘Ans. 500 men. 
S<-9. If $400, at 7%, in 9mo., produce $21 interest, what will be 
the interest on $360, for 8 mo., at 6%? Ans. $14.40. 


10. From the milk of 80 cows, each furnishing 16 qt. daily, 24. 


cheeses of 55 1b. each are made in 386 days; how many cows, 


giving 44 gal. daily, will be required, to produce, in 380 days, 33 
cheeses of 1 cwt. each ? Ans. 80 cows. 


11. How many persons can be supplied with bread 8 months, 


for $50, when flour is $5 a barrel, if, when it is $7\a‘barrel, $21. 


worth of bread will supply 6 persons’ 4 months ? Ans, 10. 
15 Dive sh 
\ 


CHAPTER: XXXII. 
ANALYSIS. 


463, Analysis, in Arithmetic, is the process of arriving 
at a required result, not by formal rules, but by tracing 
out relations and reasoning from what is known to what 
is unknown. We generally reason from the given num- 
ber to 1, and from 1 to the required number. 


The rules in this book have been in many cases deduced from exam- 
ples solved by Analysis. Analysis may also be applied to examples in 
Simple and Compound Proportion, and in Reduction of Currencies, as 
well as to a great variety of miscellaneous questions. 


we? = ANALYSIS. ips we 


Ex, 1—If 8 yd. of cloth cost $40, what will 24 yd. cost? 


This example has already been solved by Simple Proportion, p. 277. 
_ By Analysis, we should reason thus :—If 8'yd. cost $40, 1 yd. will cost 
t ace $40, or $5; and 24 yd.-will cost 24 times $5, or $120. Ans. $120. 


Ex. 2.—If 6 men can mow 30 acres of grass in 5 days, 
working 8 hours each day, how. many acres can 4 men 
mow in 9 days, of 10 hours each ? j 

This example has already been solved by Compound Proportion, p. 279. 
By Analysis, we should reason thus :— 


Af 6 men, in 5 days, working 8 hr. a day, can mow 80 acres, 
1 man, in 5 days, working 8hr.aday, “ “ 5 * 
1 man, in 1 day, working 8hr.aday, “ ‘* 1 acre i 
1 man, in 1 day, working 1 hr. a day, t 4 | 
-4men, in 1 day, working lhr.aday, “ “ 4 “ 
Aci 
oo 
2 


4 men, in 9 days, working lhr.aday, “ “-—~}> 
_ men, in 9 days, working 10 hr. a day, ‘ 45 acres. Ans. | 


en eee eee —s / ee | 
OO CLI LENCO ta tas, 


EXAMPLES FOR PRAOTION, ag oe, 


Solve the first 8 examples by both Analysis and Simple Pro- 
portion, the next 8 by both Analysis and Compound Proportion. 
1. If 12 barrels of cider cost $54, what will 15 barrels cost ? \ 
20 barrels? 100 barrels ? First ans. $67.50. 
9. How long will it take 2 men to hoe a field of corn, if 6 men 
ean do it in 7 days? . pa 
3. How many times will a wheel revolve in going 1 mi. 2 fur... \.~ 


‘ erat it revolves 12 times in going 10 rd. ? Ans. 480 times. /. 
ee 4, At the rate of $6 for 20 square feet, what will an acre of 
aoe cost ? : Ans. $18068. e 4 


% 


i 5. If a locomotive can run 40 mi. 1 fur. 20rd. in one hour . 
how far can it go in 10 minutes ? 


i “In stead of reasoning from Lhr. to 1 min., and from 1min, to 10 min., we may 
; at once, 10 min. is 7 of 1 hour ; therefore in 10 min. it ean go 2 of 40 int. i fur. 
20 rd. 


6. If % of a farm is worth $1860, what is the whole worth? 

7. A person bequeathed $4800, which was ;2, of his property, 
to charitable societies. How much was he worth ? 

_ 8. If the freight on 2 cwt. 1 qr. is 224d., at the same rate 
i. what will be the freight on 2 T. 14 ewt. ? ie £2 5a, 


Pa. ee Se 


aS ns x MY 
SB ee, a ae 
: s% a wy 4 : 
EXAMPLES IN ANALYSIS. ~ ye 283 


9. A miller had to transport 21600 bushels of grain from a 
railroad depot to his mill. In 8 days, 10 horses had removed 7200 
bushels; at this rate, how many horses would be required to re- 


move what remained, in 10 days? , Ans. 6 horses. . 
10. If 2 loads of hay will serve 3 horses 4 weeks, how many Va 
days will 5 loads serve 6 horses? 2/4, 47 Ans. 85 Ta Sees p 


11. An oblong field 8 rd. wide, 830 ft. long, contains an acre; . 
how wide is a field that is 80rd. long and contains 5 A.?) 2 
12. If the freight on 18 hhd. of sugar, each weighing 94 cwt., 
for a certain distance, costs $51.80, how much, at the same rate, 
will it cost to transport 82 hogsheads, each weighing 10} cwt., 
twice that distance ? Ans. $196. 80. 
13. How much will 46 men and 24 boys earn in 60 days, if 
the wages of 5 men for 5 days are £7 10s., and the wages of 10 
boys for 10 days are £10? Ans. £972: 
14. A garrison of 800 men have food enough to last them 60 
days, allowing each man 2lb,a day. After 20 days, a detach- 
ment of 200 men leave; how long will the remaining provisions — 
supply the men that remain ? Ans. 534 days. 
+15. A garrison of 900 men have food enough to last them 40 — 
days, allowing each man 2 Ib. a day. After 10 days, they are re- 
-inforced by 800 men, and their allowance is reduced to 141]b. a 
day ; how long will their supplies then last ? Ans. 30 days. 
“16. A body of 450 men have to march 480 miles. The first 
ten days, marching 6 hours a day, they go 1 150 ) miles; how long 
will it take them, marching 8 hours a day, at(the samo rate, to. . 
complete the distance? ,_. j Po pL pees 
17. If a farmer buys 4% cows, at $45 apiece, and pays ie them / / e 
with hay, at $18 a ton, how many tons must he give? Ans. 10. E 
"18. How many bushels of potatoes, at 80c. a bushel, yy it 
take to pay for 12 pair of hose, at 50c.? | 4) 
19. Bought some land, at $4.50 an acre; paid for it vith 
270 Cd. of wood, valued at $5 a cord. How many acres of land ~ 
were bought ? Ans.. 800 A. 
20. How much butter, at 30c. alb., will pay for 2 boxes of tea, 
containing 54 lb. each, at $1.30 a lb. ? 


Gs 
©, 


ae 


294 ANALYSIS. \/ ae 


21. A can doa piece of work in 8 days, Bin 5 days, C in 4 
days. In how many days can they do it, working together ? 


In i day, A can do }, B 3, C 4; and all three can do } + } + 3, or 42, 
they can do 27, to do $&, or the whole, will require as many dina as 47 is contained 
times in 60, or 133 days. Ans. 


22. A can mow a field in 6 days, B in. 5, C in 44, D in 3. How 
long will it take all four to do it ? Ans. 1,4 da. 

23. A, B, and ©, can clear a piece of land in 10 days; A and 
IB can do it in 16 days; how long willit take CO? Ans. 263 da. 

24. The head of a fish is }‘of its whole length; its tail is 4 of 
its length; its body is 7inches. How long is the fish? 


Head and tail together are } + 2, or 3, of the whole lengib The body, there- 
fore, is 22 — see or 7x, of the poole length. If 7 inches are ,%, 3, is 3 of T inches, or 
Linch; and 22, or the whole, is 12 times 1 inch, or 12 inches. Ans. 


25. A person, being asked his age, replied that 4 of his life had 
been passed in Baltimore, 3, of it in Richmond, and the remainder, 
which was 28 years, in New York ; how old was he? 

26. At 12 the hour and minute he of a clock are together ; 
when are they next together ? 


In the course of 12 hours, the minute hand overtakes the hour hand 11 times; 
to overtake it once, thefefore, will require 7; of 12 hours, or 17; hours. 1), hours 
past 12 will be 5 min. 5,4 sec. past 1. Ans. 


27. At what time between 5 and 6 will the hour and minute 
hands stand together? At what time between 8 and 9? At what 
time between 10 and 11? 

28. A agreed to work for B 60 days, on condition that he 
should receive $3.20 for every day he worked, and forfeit $1 for 
every day he wasidle. At the expiration of the 60 days, he re- 
ceived $129. How many days did he work? 

Had he worked every day, he would have received 60 times $3.20, or $192; 
therefore he lost by idleness $192 — $129, or $63. Livery day he was idle, he failea® 
to make $3.20 and forfeited $1, thus losing $4.20; hence, to lose $63, he must have 


been idle as many days as $4.20 is contained times in $63, or 15 days. If he was idle 
15 days, he must have worked 60 — 15, or 45 days. Ans. 


29. D contracted to work 80 days for C; he was to have $1.74 
for every day he worked, and to forfeit 60c. for every day he war 
idle. If, at the end of the time, D received $43.10, how many 


_ days was he idle? Ans, 4 sh 


Tfin 1 day 


aie. 


COLONIAL CURRENCIES, 285 


Reduction of Currencies. / 


464, Reduction of Currencies is the process of finding 
what a sum expressed in one currency is equivalent to in 
another. 

465. CotontaL CurrENCIES.—Sterling money was for- 
merly the legal currency of this country. Federal money 
took its place in 1786; but the old denominations were 
long retained, and we sometimes still hear the prices of 
articles given in shillings and pence. 

The word shilling dees not denote the same value in 
all the states. This is because the colonial paper cur- 
rency in some had depreciated more than in others; that, 
is, the colonial pound, shilling, and penny, were not worth 
so much in dollars and cents in one state as in another, 


GEORGIA In roti and "iggy Rae 81 1g, = 2130 £1 = $42 
CURRENCY. 8. Carolina, ) —= 4 . = 21¥¢, = $42 


Tn ae England, 


Virginia, Kentucky, 
and Tennessee, 


New Ena. l 
In Pennsylvania, tr 


Sgt) or. eS 2 — 21 
CURRENCY, | 6s. =$1 1s. = 163, £1=— $8} 
PENN. 


CurRENCY. pate, Jersey, Dela- 


ware, Maryland, 


7s, 6d. = $1. 1s. =13he. £1— $22 


Neetthest In New York, Ohio, 
Gana ‘ Michigan, and N, *8s. 0 = $1 1s, 124, £1 = $24 
URRENCY. is aroln a, 


Ex.—W hat will 2 dozen tumblers cost, at 9d. apiece, 
New England currency ? 


By Analysis :—In N. E. currency, 6s. or 72c. = $1; hence 9d. is 4 of 
$1. 24 tumblers, at $4 apiece, will cost 24 times $4, or $3. Ans. $3. 


EXAMPLES FOR PRACTICE. 


1. At the rate of 9s. a day, New England currency, what will 
be the wages of 4 men, for 10 days ? Ans. $60. 


i 


464. What is Reduction of Currencies ?—465. Why do we sometimes still hear 
the prices of articles named in shillings and pence? How did the word shélling 
eome to denote different values in different states? Name the different colonial cur: 
rencies, What was the value of the shilling and pound in each? 


286 ANALYSIS, 


9. At 6d. apiece, N. Y. currency, what cost 3 dozen pencils? 
8. What cost 364 yd. linen, at 7s. 6d., Penn. currency ? 
4, Reduce £42 10s., Georgia currency, to Federal money. 


Reduce £14 2s. 4d. Sum of ans. $242.64. 
5. At 9d. a yard, New England currency, what will 4 pieces 
of calico, averaging 48 yd. each, cost? Ans. $24, | 


466. Foreign Correncres.—The value of certain for- 
eign currencies in Federal Money is fixed by Act of Con- 
gress or by commercial usage, as follows :— 


VALUE oF Forricn Currencies iw U. S. Money. 


~ Banco Rix Dollar of $0.53 Millrea of Azores, $0.834 
_ Denmark, ‘ Millrea of Madeira, 1.00 
Banco Rix Dollar of Millrea of Portugal, 1.12 
Mpa and Nor-> 0.39% | Ounce of Sicily, 2.40 
Pagoda of India, 1.94 
Dollar "Thaler of Bre- 0.71. Piaster of Turkey, 0.05 

men, Pound Sterling, Gr’t l 484 
Dollar of Rome, 1.05 Britain, ete 
Ducat of Naples, 0.80 | Pound Sterling, Brit- 

Florin of Austria, Bo- 0.484 ish Provinces, Cana- 4.00 

hemia, Augsburg, : da, Nova Scotia, &c., 

Florin of Basle, 0.41 | Real Plate of Spain, 0.10 
Florin (Guilder) of Real Vellon of Spain, 0.05 

Netherlands and § 0.40 | Rix Dollar of Bremen, 0.78% 

Germany, S| Rix Dollar of Prussia 
Florin of Prussia, 0.22% and Northern Ger- > 0.69 
Franc of France and? 9 496 many, 

Belgium, “10 ) Ruble of Russia, silver, 0.75 
Guilder of Brabant, 0.383% | Rupee of British India, 0.444 
ae of Sardinia, 0.1855, | Scudo of Malta, 0.40 
Lira of Tuscany 0.16 0.99 
Livre of Genoa, 0.18,8, | Poudo of Rome, 0.994 
Livre of Leghorn, 0.16 | Specie Dollar, Denmark, 1.05 
Livre of Neufchatel, 0.264 | Specie Dollar of Swe- 1.06 
Livre of Switzerland, 0.27 den and Norway, : 
Livre Tournois, France, 0.184 | Tael of China, 1.48 
Mare Banco, Hamburg, 0.35 | Tical of Siam, ° 0.61 


Ex. 1.—Reduce 75 rix dollars of Bremen to U. 8. cur- 
rency. 
By the Table, 1 rix dollar of Bremen = §0.783. 
75 rvix dellars = 75 times $0.782, or $59.0625. ~Ans. 


REDUCTION OF CURRENCIES, 287 


Ex. 2.—Reduce $560 to millreas of Portugal. 
By the Table, $1.12 = 1 millrea of Portugal. 


$560 will equal as many millreas as $1.12 is contained times in $560, a 


ow 500. Ans. 500 millreas. 


f 


YY 


HXAMPLES FOR PRAOTICE: 


1. How many dollars equal 1000 francs ? Ans. $186. 
2. Reduce $725 to Austrian florins. Ans. 149482 fl. 
8. What is the value of 6000 Swiss livres ? Ans. $1620. 
4. How many Canada pounds are 20 eagles worth? Ans. £50. 

5. 5s. Halifax money equals how much in U. S. gold? 

6. What is the value of 16 half-eagles in ducats ? In piesers 4 
In silver rubles? In mares banco? 

7. Bought some East Indian goods for 200 rupees; what did 
they cost in Federal money ? Ans. $89. 


8. How many sovereigns (the coin that represents the pound 


sterling of Great Britain) will pay the duty on a lot of worsted hose, 
costing $1452, the rate being 35 % ad valorem ? Ans. 105 sov. 
9. Reduce 600 specie dollars of Denmark to U. S. money. 


es 


CHAPTER XX XTITf. 
EXCHANGE. 


467, Exchange is a method by which a person in one 
place makes payments in another by means of written 
orders, without the transmission of money. 


468. A Bill of Exchange, or Draft, is a written order. 


on one party to pay another a certain sum, at sight or 
some specified time. 

469. The parties to the transaction are, the Drawer, or 
Maker, who signs the bill; the Drawee, to whom it is ad- 


467. What is Exchange ?7—468, What is a Bill of Exchange ?—469, Name the 
parties to the transaction. 


yy 


288 / EXCHANGER, | 


dressed ; the Payee, to whom it is ordered to be pail; 
and the Buyer or Remitter, who buys or remits it, and 
who may be the payee or not. 


470. When a draft is presented to the drawee, if he 
acknowledges the obligation, he writes the word Accepted, 
with the date and his name, across the face of the bill, and 
thus makes himself responsible for the payment. This is 
called accepting the draft. ? 


471, As in the case of notes, three days of grace are allowed for the 
payment of drafts. But in New York, Pennsylvania, Maryland, and some 
other states, it is customary to pay sight drafts on presentation, and of 
course no acceptance is then necessary.—As regards protesting and 
the responsibility of endorsers, the same rules apply to drafts as to 
notes, § 361. 

472. Suppose Aaron Brooks, of St. Louis, owes Cobb & Doiine of 
N. Y., $1000. He buys of Eugene Ford & Co., bankers in St. Louis, a 
draft for $1000 on their correspondents, Gregory & Co., of N. Y., as fol- 
lows :— 

$1000. St. Louis, July 20, 1866. 
Ten days after sight pay to the order of Aaron Brooks one thou- 


sand doliars, value received, and charge the same to account of 
Eucenr Forp & Co. 


To Messrs. Gregory & Co., N. Y. 


Brooks endorses this draft, “ Pay to the order of Cobb & Deming,” 
affixes his signature, and remits it to the latter. They, on its receipt, 
present it to Gregory & Co., who accept it July 27th, and pay it thirteen 
days afterwards.—Here, Ford & Co. are the drawers; Gregory & Co. are 
drawees and also acceptors; Brooks is payee, endorser, and remitter ; 
Cobb & Deming are holders, as Jong as they retain the draft in legal pos- 
session. If they desire to pass it before maturity, they endorse it, and 
thus render it negotiable. 


473. When a draft costs its exact face, exchange is 
said to be at par. When it costs more than its face, ex- 
change is said to be above par, at @ premium, or against 
the place where the draft is drawn ; when less, exchange 
is below par, at a discount, or in FAR OF of the place 
where the draft is drawn. 


470, What is meant by accepting a draft o—AT. What is the custom as regards 
allowing days of grace for the payment of drafts ?—472. Give the form of a draft, il- 
lustrate:. its use in making a remittance, and name the parties concerned.—473. When 
— is exchange said to be at par? When, above par? When, below par? When 
is it against a place, and when én dis favor? 


DOMESTIC EXCHANGE. 289 


Domestic Bills of Exchange. 


474, Domestic, or Inland, Bills of Exchange (common- 
ly called Drafts) are those that are payable in the coun- 
try in which they are drawn. 

475. Operations in Domestic Exchange are similar to 
those in Stocks. : 


Ex. 1.—Bought in Louisville a thirty-day draft on 
New York for $300, at $% premium. What did it cost ? 
$1, at +% premium, cost $1 + $.0025 = $1.0025. 

$300 cost 300 times $1.0025, or $300.75. Ans. 
Ex. 2.—How large a draft on Milwaukee can a person 
in N. Y. buy for $1000, when exchange is at a discount 
of $ per cent? 


$1, at 4% discount, will cost $1 — $ .005 = $ .995. 
For $1000 can be bought a draft for as many dollars as $.995 is con- 
tained times in $1000, or $1005. 03. Ans. 


476. Rurus.—I. To find the cost of a domestic bill, 
multiply the cost of $1 at the given rate of premium or 
discount, by the face of the bill. 


If. 7 find the face of a bill that a given sum will 
buy, divide the given sum by the cost of $1. 


EXAMPLES FOR PRAOTIOER. 


1, What is the cost of a sight draft on Mobile for $1800, at 1% 
per cent. premium ? Ans. $1831.50. 

2. How large a draft on Cincinnati can a person in St. Paul 
buy for $2500, when exchange is 24 against St. Paul ? 

3. The course of exchange on Baltimore being 4% premium 
for sight, and 3% discount for sixty days, what must I pay for 
a sight draft on Baltimore for $1000 and a sixty-day draft for 
$750? Ans. $1749.875. 

4, A person living in Portland sold some property in Galveston 
for $10500. Would it be better for him to draw on Galveston for 


474. What are Domestic, or Inland, Bills of Exchange ?—475. To what are opera- 
tions in Domestic Exchange similar? Explain Exs, 1 and 2.—476. Recite the rules, 


13 


290 EXCHANGE. 


this amount and pay 24% for collection, or to have a draft on Port- 
land bought with said amount and remitted, exchange on Portland 
being at a premium of 3 per cent. ? 
Ans. Gain by drawing on Galveston, $95.83. 
5. B, living in Detroit, holds 100 shares of the Phenix Bank, 
of New York. The bank declares a dividend of 4%. 3B draws for 
his dividend, and selis the draft at 1% premium. What does he 
realize ? Ans. $4046 
| Foreign Bills of Exchange. VA 


477. Foreign Bills of Exchange are those that are 
drawn in one country and payable in another. 

478. By a Set of Exchange are meant two or more — 
bills of the same date and tenor, only one of which is to 
be paid. They are sent by different mails; and the ob- 
ject of drawing more than one is to save time in case one 
is lost. 


479, EXcHANGE oN ENGLAND.—Exchange on England 
is always at a premium in the United States, and thus the 
balance of trade always appears to be against this coun- 
try. This is because the base of computation is made the 
old value of the pound sterling, $4°, or $4.444; whereas 
the intrinsic value of the new Victoria sovereign is about 
$4.862, which is 1093 % of $4.44. When, therefore, sight 
exchange on England is quoted at 1094, or 93% premium, 
it is really at par. 

Ex. 1.—What is the cost (in gold) of the following 
foreign bill, at 94% premium ? 

Exchange for £250. Boston, July 24, 1866. 

Sixty days after sight of this Lirst of Eechange 
(Second and Third of the same date and tenor unpaid) *, 


* The Second Bill of the Set would read, “of this Second of Exchange (First and 
Third of the same date and tenor unpaid)", The Third would run, “ of this Third 
of Exchange (First and Second, &c.)”. 


477, What are Foreign Bills of Exchange ?—478. What is a Set of Exchange? 
What is the object of drawing more than one bill?—479. How does exchange on 
England always stand in the U.S.? Why is this? When is exchange on England 
really at par? Give the form ofa foreign bill of exchange. Explain Ex, 1. 


EXCHANGE ON FOREIGN COUNTRIES. 291 


pay to the order of J. M. Mosely two hundred and fifty 
pounds sterling, value received, with or without further 
advice. 
Warp & SuNDERLAND, 

Zo Hamizton Broruers, London. 

£1 = $4f, nominal par. At 944 premium, £1 costs $42 x 1.0925; 
and £250 will cost 250 times as much, or $4 x 1.0925 x 250 = 
$1218.89. Ans, 
+ Ex, 2.—For what amount will $1213.89 purchase a 
bill on London, when exchange is 109}? 


In Ex. 1 we found that, at 109}, £1 — $42 x 1.0925, or $4.853. 
Hence $1213.89 will buy a bill for as many pounds as $4.853 is con- 
tained times in $1213.89, or 250. Ans, £250. 


480. Rurus.—I. To find the cost of a bili on England 
(in gold), multiply together 34), 1 ztnereased by the pre- 
mium, and the face of the bill in pounds. 

II. Zo find the Jace of a bill that a given sum (in gold) 
will buy, divide the given sum by the product of $40 and 
1 encreased by the premium. 

In examples under Rule I., shillings and pence must be reduced to 
the decimal of a pound; and the decimal of a pound, in answers of ex- 
amples under Rule IJ., must be reduced to shillings and pence. 

481. ExcHaNGE ON OTHER CouNnTRIES.—Exchange on 
France is quoted at so many francs and centimes to the 
dollar. A franc, at par, = 18,6, cents; a centime is ;4, 
of a franc. 

Exchange on other countries is quoted at so many 
cents to some coin taken as a standard: thus, on Ham- 
burg, 354 cents to the mare banco; on Amsterdam, 39 
cents to the florin, &c. 

Jn these cases, the cost of a bill, and the face of a bill 
that a given sum will buy, are readily found by Analysis, 
as in Reduction of Currencies. 


Explain Ex. 2.480. Recite the rule for finding the cost of a bill on England, 
Recite the rule for finding the face of a bill thas a given sum will buy. What reduc- 
tions must be made ?—481. How is exchange on France quoted? How is exchange 
on other countries quoted? In these cases, how are the cost of a bill, and the face 
of a bill that a given sum will buy, found? Explain Ex. 3. 


292 EXCHANGE. 


Ex. 3.—What is the value of a bill on Havre for 1206 
francs, exchange being 5 francs 18 centimes to the dollar? 


If 5 francs 18 centimes = $1, a bill for 1200 francs will cost as many 
dollars as 5.18 is contained times in 1200, or 231.66. Ans. $231.66. 


EXAMPLES FOR PRAOTIOCE. 


1. What is the cost, in gold, in N. Y., of a set of exchange on 
Dublin for £450 10s., at 92% premium ? Ans.. $2197.44. 

2. What is the cost, in gold, of a bill on Paris for 7500 francs, 
when exchange is 5 fr. 10 cen. to the dollar ? Ans. $1470.59. 

3. When the course of exchange is 75jc. to the ruble, what 
will a bill on St. Petersburg for 2400 rubles cost ? 

4, How large a bill on Bremen can be bought for $2000, when 

exchange is 79c. to the rix dollar ? 

5. Exchange on Liverpool standing at 109, what will a bill on 
that city for £1500 2s. 6d. cost ? Ans. $7267.27. 

6. A New York merchant, owing a debt in London, can pur- 
chase gold at 145, and with it buy exchange at 94% premium; or 
can remit U.S. 10-40’s, and sell the same in London at 604. How 
low must he buy the bonds (for currency), to make a saving by 
remitting them in stead of a bill of exchange ? 


Each $1 of bonds transmitted would be worth $ .605 x 1.095, in gold. Reducing 
this value to a currency basis, we have $.605 x 1.095 x 1.45 = $.96+. If, therefore, 
the bonds can be bought for less than 96, there will be a saving in remitting them. 


Arbitration of Exchange. 


482. Arbitration of Exchange is the process of finding 
the rate of exchange between two countries, when there 
have been intermediate exchanges through other coun- 
tries. In Arbitration, we use what is called Conjoined 
Proportion or the Chain Rule. 


A merchant, for example, may remit from New York to Hamburg, by 
remitting from New York to London, from London to Paris, from Paris 
to Amsterdam, and from Amsterdam to Hamburg. The rate of this 
Circuitous Exchange, as it is called, will probably differ somewhat from 
that of a direct remittance from New York to Hamburg; to find 
whether it will cost more or less, is the object of Arbitration. 


482. What is Arbitration of Exchange? Give an illustration of Circuitous Ex- 
ehange.—483, Recite the Chain Rule. 


ARBITRATION OF EXCHANGE. 293 


483, Cuan Rute.—1. Write the equivalents by pairs, 
each with its denomination, on opposite sides of a vertical 
line, commencing on the left with the denomination of the 
required sum, and on the right with the given sum to be 
remitted; and arranging the terms so that each denomi- 
nation on the right may correspond with the one neat be- 
low tt on the left. 

2. Cancel common factors on the left and right, and 
divide the product of the remaining terms on the right by 
that of the remaining terms. on the lefe. 

If the terms are properly arranged, the last denomination on the 
right will correspond with the first on the left. - 

Ex.—When exchange at New York on London is at 
10% premium, at London on Paris 27 francs 20 centimes 
to £1, at Paris on Amsterdam 9 stivers to 1 franc, and at 
Amsterdam on Hamburg 18 stivers to 1 mare banco, what 
will it cost to remit 5000 mares banco from N. Y. to Ham- 
burg, through London, Paris, and Amsterdam? Would 
it be better to remit in this way, or direct from N. Y. to 
Hamburg, the rate being 36 cents to the mare banco— 
and how much ? 


$2 5000 mares b. Cancelling: 6606 625 
1 marc b. | 18 stivers $4 21.2118 2 
9 stivers | 1 franc 1.7 9 | 44 
27.2 fr. aL ‘ 
£9 | $40 x 1.10 625 x 44 = 27500 
1:7.X%.9 = 15.3 


27500 +- 15.8 = $1797.39 
Direct Exchange, $0.86 x 5000 = $1800.00 
Circuitous Exchange, 1797.39 


Gain by Circuitous Exchange, $2.61 Ans, 


In this example, £1— $42 x 1.10; hence £9 = $40 x 1.10, as given 
above.—The relative value of different measures, weights, and goods, may 
be found, on the same principle, by the Chain Rule. 


EXAMPLES FOR PRAOTIOE. 


1. A person in Philadelphia desires to pay £1800 in Liverpool. 


Exchange on Liverpool is at 92% premium, on Paris 5 francs 15 
J 


294 . EXCHANGE, 


centimes to a dollar. Exchange on Liverpool in Paris is 25 franes 
15 centimes to the pound sterling. Is it better for him to remit 
direct to Liverpool, or through Paris, and how much ? 
_ Ans. Gain by direct remittance, $10.29. 
2. A New York merchant orders £1000 duo him in London 
to be remitted by the following route: to Hamburg, the course 
of exchange being 14 marcs banco to the pound; thence to Co- 
penhagen, at 14 mares banco to the rix dollar; thence to Bor- 
deaux, at 2 francs 80 centimes to the rix dollar; thence to N. Y., at 
5 francs 30 centimes to the dollar. How many dollars did he re- 
ceive ? Ans. $4980.82. 
Would he have gained or lost by drawing directly for the 
amount on London, and selling his draft at 1094, leaving interest 
out of account ? 


8. If 16 barrels of cider are worth 64 bushels of corn, and 15 
bu. of corn are worth 2 barrels of flour, and 3 tons of coal are 
worth 4 barrels of flour, and 16 Ib. of tea are worth 2 tons of coal, 


lew many pounds of tea are equal in value to 7 barrels of cider? 
Ans, 222 Ib,: 


CHAPTER XXXITII. 


PARTNERSHIP. 


484, A Partnership is a business association between 
two or more persons, who agree to share the profits or 
losses. Persons so associated are called Partners, 

Capital is money invested in business. 


Different agreements are made between partners as to the division of 
profits. One may contribute the capital, and another his services, and 
they may divide equally. Or all may contribute capital and labor equally, 
and make an equal division. When different amounts of capital are fur. 
nished, and little or no labor is required, or all contribute equally of 


484, What isa Partnership? What is Capital? What is said about the divis- 
jon of profits among partners ? 


PARTNERSHIP. 295 


their labor, the profit or loss is usually divided according to the amounts 
of capital furnished. 

485. Case I.—To find each partner's share, when they 
Surnish capital for the same length of time. 

Ex. 1.—A, B, and C, engaged in a speculation. A 
put in $180, B $240, C $480. They gained $300; what 
was each partner’s share ? 


The whole capital employed was $180 + $240 + $480, or $900. Since 
$900 capital gained $300, $1 of capital gained 545 of $300; and A’s capi- 
tal of $180 was entitled to $§$, B’s $240 to $$$, and C’s $480 to S00, of 
$300. The operation is proved by adding the shares found, and seeing 
whether their sum equals the whole gain. 


A’s capital, $180 A’s share, 4 soo oo of $300 = $60 
Bisr. 240 B’s Gap. °° == 80> Ans. 
C’s (79 480 C’s oe $33 (79 (79 —_ 160 

Total capital, $900 Proor: Gain, $300 


Rutze.—Make each pariner’s capital the numerator of 
a fraction, and the total capital the denominator ; for 
each partner's share, take his fraction, thus formed, of 
the whole gain or loss. 

Ex. 2,.—T'wo brothers, the one 18 years old and the 
other 21, contribute $468 for the support of a parent, in 
the ratio of their ages. What, does each give ? 


This example is analogous Ist 18 48 of $468 — $216 Wh: 
to a question in Partnership. 2d 21 4% of $468 = $252 tee 
There are in all 18 + 21, or 39, 39 Proor: $468 


parts; of, which one furnishes 
18, the other 21. 


EXAMPLES FOR PRAOTIOE. 


1. The profits of Mason, Dean, & Co., for one year, are $9275. 
Mason contributes $20000 capital; Dean, $12500; and Graham 
(who is the Co.), $4600. What is each partner’s share of the 
profits? Ans. Mason’s, $5000; Dean’s, $3125; Graham’s, $1150. 

2. A and B buy a house for $2500, A furnishing $1200, B 
$1300. They receive $210 rent; how should it be divided ? 


3. Ames, Boorman, & Crane, buy a hotel for $18500, of which 


485, What is Case 1.? Explain Ex.1. Recite the rule. Explain Ex. 2 


296 PARTNERSHIP. 


Ames contributes $8000, Boorman $6200, and Crane the rest. 
They sell it for $16975, and their expenses are $825. How much 
of the loss must each bear? Ans. A., $800; B., $620; C., $430. 

4. Two persons hire a pasture for $30. The first turns in 8 
cows; the second, 5. How much ought each to pay ? 

5, A, B, OC, D, and E, are to divide $2400 among themselves. 
A is to have 4, B 4, C 3; D and E are to divide the remainder in 
the ratio of 5 to 7. How much should each receive ? 

Last answers: D, $208.3384; E, $291.663. 

6. A person wills to his elder son $1200, to his younger $1000, 
to his daughter $600. But it is found that his whole property is 
worth only $800. How much should each receive ? 

7. X, Y, and Z, embark in a speculation, X furnishing $ the 
capital, Y 2 of the remainder, and Z the rest. Their profit is 
$1900, and X is allowed $100 for attending to the business. How 
much does each receive ? Ans. X, $1000; Y, $600; Z, $300. 


486. Casr Il.—70 find each pariner’s share, when they 
furnish capital for different lengths of time. 

Ex. 1.—Three partners, O, P, and Q, furnished capital 
as follows:-O put in $400 for 2mo.; P, $300 for 4 mo. ; 
Q, $500 for 3mo. They gained $350; what was the share 


of each ? 
O’s $400 for 2mo. = $800 for 1 mo. 
P’s $300 ** 4mo. = $1200 “ Imo. 
Q’s $500 “ 8mo. = $1500 “ Imo. 


The whole capital is therefore equivalent to $3500 for 1 month; ard, 
as O put in what is equivalent to $800 for 1 mo., he is entitled to 89,0; 
of $850, or $80. In like manner, P is entitled to $2$5 of $350, or $120; 
and Q, to 4289 of $350, or $150. 

Ruie.— Multiply each partner's capital by tts time. 
Treat this product as his capital, and proceed as in Case I. 

Ex. 2.—Three partners were in business for 12 months, 
and cleared $2919. The first had $4000 in the whole 
time. The second put in $5000 three months after the 
partnership commenced, and three months afterwards 
$3000 more. The third put in $3000 on starting, but with- 


486. What is Case II.? Explain Ex.1. Recite the rule. Explain Ex, 2 


PARTNERSHIP. 297 


drew $2000 four months before the partnership expired. — 
Divide the profit. 
1st $4000 x 12 = 48000 Share, ;48; of $2919 = $1008, 


2d $5000 x 9 = 45000 
$3000 x 6 = 18000 


GS000 x-see Saree dte “Hey an they ix, $1993. 
8d $3000 x 8 = 24000 
$1000 x 4= 4000 
28000  auSharey Aig rcety exe G88. 
139000 Proor: $2919. 


EXAMPLES FOR PRAOTIOCE. 


1. A and B enter into partnership, A furnishing $325 for 6 
months, and B $200 for 8 months. There is a loss of $100; 
what is the share of each ? Ans. A, $54.93; B, 45.07. 

2. Two partners received $300 for constructing a piece of 
road. The first furnished 5 laborers for 9 days; the second, 7 
laborers for 11 days. What was the share of each ? : 

3. Three farmers hired a pasture for $55.50. The first put in 
6 cows for 3mo.; the second, 8 cows for 2mo.; the third, 10 
cows for 4mo. "What must each pay ? 

4. For the transportation of some flour 93 miles, I have to 
pay $116.25. A carried 50 bar. 70 miles; B, 10 bar. 98 miles; OC, 
40 bar. 53 miles; D, 50 bar. 23 miles; E, 40 bar. 40 miles. How 
much must I pay each ? Ans. A, $43.75, &e. 

5. A, B, and OC, began business Jan. 1 with $650, furnished by 
A; April 1, B put in $500; July 1, O put in $450. The profit 
for the year was $375; divide it. Ans. A, $195, &e. 

6. D, E, and F, were interested in a coal mine, and cleared 
the first year $8285. D had $10000 invested for 9 mo., when he 
withdrew half of that sum; E put in $20000, 2 mo. after the part- 
nership was formed; and F put in $12000, 5 mo. before it expired. 
_ Divide the profit. Ans, D, $945; E, $1800; F, $540. 

7. Two partners, G and H, cleared in 6 mo. $2150. G’s capi- 
tal at first was to H’s as 2 to 1. After 2 months, G withdrew 4 
of his capital, and H 4 of his, Divide the profit. 

Ans. G, $1400; H, $750.. 


298 ALLIGATION. 


CHAPTER XXXIV. 
ALLIGATION. 


487. Alligation is the process of solving questions as 
to the mixing of ingredients of different values. There 
are two kinds of Alligation, Medial and Alternate. 


Alligation means connecting, and the process is so called from ¢on- 
necling or linking the prices of the ingredients together, as shown in $ 490. 
Alligation Medial. 


488. Alligation Medial is the process of finding the 
average value of a mixture, when the value and quantity 
of each ingredient are known. 

Ex. 1.—A grocer mixes 70 1b. of tea worth $1 a |b., 
100 lb. worth $1.25, and 30 1b. worth $1.50. What is a 
pound of the mixture worth ? 


70 Ib., at $1, are worth $70; 100 Ib., at rocx dete 20 
$1.25, are worth $125; 30 1b., at $1.50, are 100 x 1.25 a 125 
worth $45. The whole mixture, therefore, is eer ri aay 5 
worth $70 + $125 + $45, or $240; and it con- _30 x 1.50 = 40 
tains 70 + 100 + 30 1b., or 200 lb. If 200 1b. 200) 240 
are worth $240, 1 lb. is worth gbp of $240, or Ans. $1.20 
$1.20. Ans. i ; 


489. Rurz.— Divide the total value of the ingredients 
by the sum of the quantities. 


If an ingredient is put in that costs nothing (as water, chaff), its 
quantity must be added in with the rest, though its value is 0. 

The principle of this rule applies to many questions that involve the 
finding of an average, besides those relating to values or prices. 


EXAMPLES FOR PRAOTIOER. 


1, A liquor-merchant mixes 32 gal. of wine at $1.60 a gallon, 
15 gal. at $2.40, 45 gal. at $1.92, and 8 gal. at $6.80. What is the 
value of a gallon of the mixture ? Ans. $2.008. 
2. If a ship sails 5 knots an hour for 8 hours, 7 knots for 5 
hours, and 8 knots for 4 hours, what is her average rate per hour ? 


487. What is Alligation? Name the two kinds of Alligation, Why is the pro- 
oess £0 called ?—488. What is Alligation Medial? Explain Ex.1. Recite the rule 


ALLIGATION MEDIAL. 299 


3. A dishonest grocer mixed 3 1b. of sand with 10 lb. of sugar 
worth 12c¢., 201b. worth 14c., and 301b. worth 16c. What did 
the mixture cost him per pound ? Ans, 1384¢. 

4, A goldsmith melts together 11 0z. of gold 23 carats fine, 
8 oz. 21 carats fine, 10 oz. of pure gold, and 2 1b. of alloy. How 
many carats fine is the mixture? Ans.’ 1228 carats. 

A carat is 3,; that is, gold 21 carats fine is 31 pure metal. 

5. If 4 dozen eggs are bought at 18% cents a dozen, 6 dozen at 
21 cents, 34 dozen at 24c., and 54 dozen at 25c., what is the aver- 
age cost per dozen ? Ans. 22,%c. 

6. A dairyman owning 30 cows finds, at a certain milking, that 
6 give 12 qt. each, 8 give 104 qt., 10 give 94 qt., and the rest 8 qt. 
apiece. What is the average? 

7. Ifa farmer mixes 10 bu. of corn, worth 80 cents a bushel, 
20 bu. worth 85c., 25 bu. worth 90c., and 20 bu. worth 95c., what 
is the mixture worth per bushel ? Ans. 88%c. 


Alligation Alternate. 


490. Alligation Alternate is the process of finding the 
quantities to be taken of two or more ingredients, of 
given values, to make a mixture of given value. 

Ex. 1.—In what relative quantities must coffees worth 
15, 16, 20, and 21 cents a pound, be Basen to make a 
mixture worth 19 cents a pound ? 


‘It is clear that the gains and losses on the several ingredients, as 
compared with the mean value, must balance. Hence we consider a 
price less than the mean with one greater,—l5c. with 2lc. On every 
pound put in at 15c. and sold in the mixture for 19c., there is a gain of 
4c.; and on every pound put in at 2lc. and sold for 19c., there is a loss 
of 2c. Therefore, as the gain and loss on equal quantities of these two 
kinds are as 4 to 2, we must take quantities that are to each other as 2 
to 4. In like manner, comparing 1b. at 16c., and 1 Ib. at 20c., we find. 
that there is a gain of 30. against a loss of Ic. ; hence the quantities taken 
must be as 1 to 3. - The relative quantities, ‘therefore, are 2 1b. at 15c¢., 
1 Ib. at 16c., 3 Ib. at 20c., and 4 Ib. at 21c. Ans. 


The brief mode of performing this operation 15 
is to link the values in pairs, one less-than the 16 
mean with one greater, to take the difference be- 19 20 1 
tween the mean and each value, and write it oppo- 21 
site the value with which it is linked. 


H> OD kt bo 


300. ALLIGATION ALTERNATE, 


The terms may be linked differently, provided a 9 
one less than the mean is connected with one 19 ie 4 
greater ; the answers, of course, differ, according to 
the linking. As these answers show merely the rela- 21 3 
tive quantities, we may multiply or divide the numbers by any common 
multiplier or divisor, and thus produce an infinite variety of answers. 


Alligation Alternate is proved by Alligation Medial. Thus :— 


Proof of 1st answer. Proof of 2d answer. 
. 21b., at 15e. = 380c. 1 Ib, at 15c. = 15c. 
RS CY ee Ut Be et A OG, heme in 
Or 2UGe es DUES SEN PIO == CUE. 
HM. GS Dc 84e, Si. HeBle” =| ape 


10 Ib. cost $1.90, or 1lb.19c. 101b. cost $1.90, or 1 Ib. 19. 


Ex. 2,—A grocer, having 10 Ib. of coffee worth 15c. a 
pound, wishes to mix it with other kinds worth 16, 20, 
and 21¢., to make a mixture worth 19c. a pound. How 
many pounds of each must he take ? 

In Ex. 1, we found the relative quantities of these coffees for a mix- 
ture worth 19 cents to be 2, 1, 8, 4, or 1, 2, 4, 3. 


Looking at the first answer, we find that the ratio of 10, the 
given quantity of 15- 
26x 5 = 10Ib. cent coffee to 2, its 1x 10 =10]b. 
1x5= 5)b. difference, is 5; there- 2 x 10 = 20]b. 4 
8 x5 = 151b. (4”% fore we multiply the 4 x 10 = 401b. 
yO ee numbers throughout 3 xy 19 — 30]b. 
by 5. 
In the 2d answer, the ratio is 10 to 1; therefore we multiply by 10. 
Ex. 3.—A grocer, having coffees worth respectively 
15, 16, 20, and 21 cents, wishes to make with them a 
mixture ef 801b., worth 19c. a pound. How many 
pounds of each kind must he use ? 


In Ex. 1, we found the relative quantities to be 2, 1, 3, 41b., or 
1, 2, 4, 8 b.,—in either 


&®xS8=—16)b. ease making a total of 1x8= Blb. 
1x 8 = 8b. 101lb. Fora mixture 9 x 8=16)]b. 
8x 8 — 24)b. Ans. of 80Ib., therefore, he 4 x 8 = 33 ]b Ans, 
4.x 8 = 32)b. must take 8 times as 3x 8 = 24]b 


much of each. 


490. What is Alligation Alternate? Explain Ex. 1. What is the brief mode of 
performing the operation? How may different answers be obtained? How is Alli- 
gation Alternate proved? Explain Ex, 2, Explain Ex 3. Recite the rule. 


ALLIGATION ALTERNATE. . 801 


491. Rutz.—l. Write the values in a column, and 
the mean value on the left. Link each value less than 
the mean with one greater, and each greater with one less. 
Write the difference between the mean and each value, op- 
posite the value it ts linked with. These differences are 
the. relative quantities of the ingredients taken in the 
order in which their values stand. 

2. If the quantity of one ingredient is given, to find the 
corresponding quantities of the others, multiply their dif- 
Serences by the ratio of the given quantity to the differ- — 
ence of the ingredient it represents. 

3. If the quantity of the mixture is given, to find the 
quantity of the ingredients, multiply their differences by the 
ratio of the given quantity to the sum of the differences. 


Ex. 4. A liquor-dealer wishes to mix three kinds of 
whiskey worth respectively $3.25, $3.50, and $3.75 a gal- 
ton, with water, so as to make a mixture worth $3. What 
parts of each must he take ? 


We represent the water by 0. As there are three values greater than 
the mean and but one less, we have to link the three with the one. There 
will, therefore, be three 

differences opposite the 


eG S21D BY | 
8 iB 3 0, and their sum will rep- 
3.4 3. ; resent the relative quan- 


3.20 3. : 
‘st tity of water. Ans. 38 gal. 
0 T5+.5+.25 = 1.5 of’ cach kind of whiskey, 
and 14 gal. of water. 


EXAMPLES FOR PRAOTIOE. 


1. In what proportions must gold, 12, 16, 17, and 22 carats 
fine, be taken, to make a compound 18 carats fine? 20 carats fine? 
164 carats fine ? First ans. 4, 4, 4, 9. 

2. A merchant wishes to mix 90 Ib. of sugar, worth 104c., with 
three other kinds, worth 10, 12, and 14 cents, respectively. How 
many pounds of each must he use, that the compound may be 
worth lle.? 124¢.? 18¢.? 

Get aan | 270 Ib. at 10c.; 45 1b. at 12c.; 901b/ at 14¢. 
Or, 801b. at 10c.; 301b, at 12¢.; 15 Ib. at 14. 


802 ALLIGATION ALTERNATE. 


3. How many pounds of spices, worth respectively 30, 40, and 
50c. a pound, must be mixed with 201b. worth 80c. a pound, to 
form a mixture worth 60c. a pound? Worth 75 cents? Worth 
45 cents ? First ans. 6% \b. of each. 

4, A man having 40 bu. of oats that cost him $22, wishes to 
mix them with two other kinds worth respectively 50 and 65c. 
How much of each kind must he take, to form a mixture worth 
60c. a bushel ? Ans. 40 bu. at 50c.; 120 bu. at 65c. 

5. B, having a contract to furnish 442 Ib. of tea worth $1.40 a 
\b., wishes to make a mixture, of the required value, out of four 
kinds, worth respectively $1, $1.10, $1.45, and $1.50. How many 
pounds of each must he take ? 

Ans. 52 1b. at $1, 26 1b. at $1.10, 156 1b. at $1.45, 208 Ib. at 81. 50. 

6. In what proportions must water, and two kinds of rum 
worth $24 and $3 a gallon, be mixed, to form a compound of 40 
gallons, worth $2 a gallon? 

7. A news-agent sold 198 newspapers, at an average price Of 
7 cents apiece. How many must he have sold at 38c., 4c., 5c., 6c., 
and 10c. ? Ans. 277 at 8c., &e. 


CuiA Del ER. XX ave 


INVOLUTION. 


492, Involution is the process of multiplying a number 
by itself. The product is called a Power of the number 
multiplied. 

2x22 4. This process is Involution; 4 is a power of 2. 

493. Powers are distinguished as First, Second, Third, 
Fourth, &c., according to the times that the given num- 
ber is taken as a factor. 

They are indicated by a figure, called an Index (plural, 
indices) or Exponent, placed above the number at the 
Sriphtis as, -2 2". 


492. What is Involution? What is the preduct obtained by Involution called? 
Give an example.—493. How are powers distinguished? How are they indicated ? 


INVOLUTION. 303 


494, The First Power is the number itself; its index 
is never written. ‘The Second Power is also called the 
Square, and the Third Power the Cube. 


First power of 2, 2 

Second power, or Square, 27=2x2=4 

Third power, or Cube, Verne eS, § 

Fourth power, Pies. Gene) hea 2 GL 


495. Rutzu.—TZo involve a number, multiply it by it- 
self as many times, less 1, as there are units in the index 
of the power required. 

3x 38=9. There is one multiplication, though 3 is used as a factor 
twice, and 9 9 ts the second power. 


496. In stead of multiplying by the original number each time, powers 
already found may be used as multipliers. Thus, for the 7th power, the 
4th may be multiplied by the 8d. But observe that the resulting power 
will be that denoted by the sum of the indices of the multipliers, not their 
PRODUCT. 


EXAMPLES FOR PRACTIOE. 


1. Give the squares and cubes of the numbers from 1 to 12. 

2. Find the 4th power of 2. Of4.6. Ans. 51%;; 447.7456. 
3. Square 89. Cube 221. Involve. 25 to the 5th power. 

_ 4, Find the value of the following indicated powers:—3°; 14'; 
5 7.2"; .O1f; @*; @t; 1°. Sum of ans. 71966,28747501, 


CAPT BY XX NE Vor 
EVOLUTION. 


497. Evolution is the process of resolving a number 
into two or more equal factors. One of these equal fac- 
tors is called a Root of the number resolved. 


4 = 2x2. This process is Evolution; 2 is a root of 4. 


494. What is the First Power of a number? What is the Second Power also 
called? The Third Power ?—495. Recite the rule for Involution.—496. What may 
be done, in finding the higher powers? What caution is given ?—497. What is 
Evolution ? 


004 EVOLUTION. 


Evolution is the opposite of Involution. In the latter, a root is given 
and a power required ; in the former, a power is given and a root required. 


498. Roots take their names, Square Root, Cube Root, 
Fourth Root, Fifth Root, &c., from those of the corre- 
sponding powers. 

Roots are indicated by a character called the Radical 
Sign, ./, placed before the number whose root is to be 
extracted. | 

The Index of a root is a figure placed above the radi- 
cal sign at the left, to denote what root is to be taken,— 
that is, into how many equal factors the number is to be 
resolved. ‘To express the square root, the radical sign is 
used without any index. : 


J 4, read square root of 4, as 
Vf 8, “ cube root of 8, Phin casa eat tage eee 
RAG, > OUTTA TOOLOT 1G, tae, ea a) Ae eee Pee 


The most important operations in Evolution are the 
extraction of the Square and the Cube Root. 


2, since 2 x. 2 = 4. 
2 


Hl Hl dl 


Square Reot. 


499. Extracting the square root of a number is resolv- 
ing it into ¢wo equal factors; as, 4 = 2.x 2. 

500. Taking the smallest and the greatest number that 
can be expressed by one figure, by two, three, and four 
figures, let us see how the number of figures they contain 
compares with the number of figures in their squares :— 
Roots, 2b 9 | 10 99 | 100 999 1000 9999 
Squares, 1 81 | 1/00 98'01 | 1/0000 99'80’01 | 1'00'00’00 99'98'00'01 

We find from these examples that, if we separate a 
square into periods of two figures cach, commencing at 
the right, there will be as many figures in the square root 
as there are periods in the square,—counting the left-hand 
Jigure, Uf there is but one, as a period. 

Of what is Evolution the opposite ?—498. From what do roots take their 
names? How are roots indicated? What is the Index of a root? How is the 
square root expressed? What are the most important operations in Evolution ?— 


499. What is meant by extracting the square root of a number ?—500. How can we 
- find, from a square, the number of figures its square root contains? 


SQUARE ROOT. 305 


501. We derive the method of extracting the square 
root from the opposite operation of squaring. Square 
25, regarding it as composed of 2 tens (20) and 5 units. 


25 = 20 +5 207 —= 400° - 25 

20 +5 20 x 5=100 25 

Multiplying by 20, 207 + (20 x 5) 2x 100=200 125 
Multiplying by 5, (20 x 5) + 5? “HE 2b 850 


Adding partial products, 20? + 2 (20 x 5) + 5? = 25 squared = 625 626 
Hence, Zhe square of a number composed of tens and 
units, equals the square of the tens, plus twice the product 
of the tens and units, plus the square of the units. 
502. Now reverse the process. Find the sq. root of 625. 


According to § 500, we separate 625 into periods of two figures each, 
beginning at the right (6'25), and find that the root will contain two 
figures,—a tens’ and a units’ figure. 

According to $ 501, 625 must equal the square of the tens in its root, 
plus twice the product of the tens and units, plus the square of the units. 
The square of the tens must be found in the left-hand period, 6(00).. The 
greatest number whose square is less than 6 is 2, which we place on the 
right as the tens’ figure of the root. 2 tens 6/95 (95 
(20) squared = 4 hundreds, which we sub- 5 (25 
tract from the 6 hundreds. Bringing down ain 
the remaining period, we have 225, which 20 x 2 = 40) 225 
must equal twice the product of the tens (40 x 5) + 57 = 295 
and units, plus the square of the units. | han 

Hence, to find the units’ figure of the root, we divide 225 by twice 
the tens, or 40. The quotient is 5, which we place in the root as its 
units’ figure. Then, twice the product of the tens and units, plus the 
square of the units = twice 20 x 5, plus 25 = 225. Placing this under 
the dividend 225, and subtracting, we have no remainder. 

6'25 (25 In practice, we write twice the tens’ figure (4) on 

the left as a trial divisor, and complete it by annexing 

45 995 the units’ figure of the root. Multiplying this com- 

22 plete divisor by the units’ figure, we have the same 
225 result, 225. 


503. Rutu.—1. Separate the given number into periods 
of two figures each, beginning at the units’ place. 

2. Find the greatest number whose square is contained 
in the left-hand period, and place tt on the right as the first 


501. Whence do we derive the method of extracting the square root? Squara 
25, regarding it as composed of 2 tens and 5 units. What principle is dedueed from 
shis example ?—502. Reverse the process; extract the square root of 625, explain- 
ing the steps.—508, Recite the rule for the extraction of the square root, 


306 EVOLUTION, 


root figure. Subtract its square from the first period, and 
to the remainder annex the second period for a dividend. 


3. Double the root already found, and, placing it on 
the left as a trial divisor, find how many times it is con- 
tained in the dividend with its last figure omitted. Annex 
the quotient to the root already found and to the trial di- 
visor. Multiply the divisor thus completed by the last root 
Jigure, subtract, and bring down the next period as before. 


4. To the last complete divisor add the last root figure 
for a new trial divisor, and proceed as before till the 
periods are exhausted. 


If any trial divisor is not contained in the dividend with its last figure 
omitted, annex 0 to the root already found and to the trial divisor, bring 
down the next period, and find how many times it is then contained. 

If, on multiplying a complete divisor by the last root figure, the 
product is greater than the dividend, the last root figure must be dimin- 
ished, and the figure annexed to the trial divisor changed accordingly. 

If, when all the periods have been brought down, there is still a 
remainder, periods of decimal ciphers may be supplied and the operation 
continued. The root figures corresponding to the decimal periods will 
be decimals. 

504. To point off a decimal for the Py 9 Find the square 


extraction of the square root, commence 
at the decimal point and go to the right, Toot of 1524,1216. 


completing the last period, if necessary, 15'24.12'16 (89.04 
by annexing a cipher. Root figures 9 

resulting from decimal periods are always —~ 

decimals. 69 624 


621 
505, To find the root of a common hs — 
fraction, reduce it to its lowest terms, (804 31216 
and extract the root of its numerator and 31216 
denominator separately, if they have ex- 
act roots. If not, reduce the fraction to a decimal, and extract its root, 
carrying the operation as far as may be required. Reduce a mixed num- 

ber to an improper fraction, and proceed as just directed. 


506. To prove the operation, square the root found, and see whether 
the result equals the given number. 


If any trial divisor is not contained in the dividend with its last figure omitted, 
what must be done? Under what circumstances must a roct figure be diminished ? 
Tf, when all the periods have been brought down, there is still a remainder, what 
may be done ?—504. How is a decimal pointed off for the extraction of the square 
roct? What root figures are always decimals?—505. How is the square root of 
a cammon fraction found? Howis the square root of a mixed number found ?—How 
is the operation proved ? 


SQUARE ROOT. 307 


EXAMPLES FOR PRAOTICE, 


1. What is the square root of 100180081 ? Ans. 10009. 
2. What is the square root of 12321? Of 53824? Of 11390625? 
Of 16064064? Of 3600? Sum of ans. 7786. 
3. Find the square root of 4489. Of 531441. Of 16983563041. 
Of 11019960576. Of 61917364224, . Sum of ans. 484925. 
4. Extract the square root of 6.5536. Of .00390625. Of 
0011943936. Of 60.481729. Sum of ans. 10.43406. 
5. Find the square root of #38. Of 434. Of4}. Of4%. Of 
49425. Of 448. Ans. 8, 24, 2.027 +, &e. 


APPLICATIONS OF SQUARE Root. 


507. The areas (§ 252) of similar figures are to each 
other as the squares of their like dimensions. The areas 
of two circles whose diameters are 3 and 4 feet, are to 
each other as 3? to 4”, or 9 to 16. 

508. When the area of a square is known, extract its 
square root, to find one of the sides. The answer will be 
in the denomination of linear measure that corresponds 
to the denomination of the area. A square field containing 
49 square rods will be 7 (/49) Linear rods on each side. 


509. A Rectangle is a four-sided figure = F 
whose angles are all right angles; as, EF 
G H. HH G 


510, A Triangle is a figure bounded by three straigh 
lines. 

511. A Right-angled Triangle is a triangle 
that contains a right angle; as, ABC. ¢ | 

The Hypothenuse of a right-angled trian- ae 
ele is the side opposite the right angle; as, 
AC. Of the two shorter sides, the one on which the 
triangle stands (as AB) is called the Base, and the other 
(as B C) the Perpendicular. 

507. What principle is laid down respecting the areas of similar figures ?—508, 
How is the side of a square found from its area ?—509. What is a Rectangle ?—510. 


What is a Triangle?—511. What is a Right-angled Triangle? What is the Hypoth- 
cnuse of a right-angled triangle? What is the Base? What is the Perpendicular? 


A B 


808 EVOLUTION. 


512, It is shown, in Geometry, that the square on the 
hypothenuse equals the sum of the 
squares on the other two sides, 


This principle is illustrated by the fig- 
ure on the right. The small squares are 
all equal; it will be seen that the square 
of the hypothenuse contains 25, that of 
the base 16, that of the perpendicular 9. 
25 = 16+ 9. Hence these 


_ Rores.—I The two shorter 
sides being given, to find the hy- 
pothenuse, add their squares and 
extract the square root of the sum. 

Il. Lhe hypothenuse and one of the shorter sides being 
given, to find the other, subtract the square of the given 
side from that of the hypothenuse, and extract the square 
root of the remainder. 

Ex.—A liberty pole was broken 30 feet from the top, 
and the upper piece, falling over, struck the ground 18 
ft. from the lower extremity. How high was the pole ? 


A right-angled triangle was formed, the A 
broken part being the hypothenuse, tne upright 30°?— 18? = 576 


part the perpendicular, and the distance from of 576 = 24 
the point where the top struck the ground to 24 + 380 = d4 
the foot of the pole the base. Applying Rule Ang Ba fe 


II., we find the perpendicular, or upright piece ; 
which, added to the part broken off, gives the whole length. 


EXAMPLES FOR PRAOTIOER. 


1, A flag-staff 36 ft. high was broken 4 of the way up. How 
far from its foot did the top strike the ground Ans, 20.7 ft. + 

2. If a ladder 35 ft. long is placed 21 ft. from the base of a 
rock, how high up the rock will it reach ? Ans. 28 ft. 

3. A rope 45 ft. long, attached to the top of a house, extended 
to a log 36 ft. from its base. How high was the house? 

4, Two persons start from the same place, and go, the one due 

512. What does the square on the hypothenuse equal? How is this principle 


illustrated? Recite the rule for finding the hypothenuse. Reeite the rule for find- 
tag the base or perpendicular. Explain the example. 


CUBE ROOT. $09 


north 80 miles, the other due west 60 miles. How far apart are 

they ? 
5, What is the side of a square whose area is 121 square fect. 
6. What is the distance between two opposite corners of a lot 


50 feet by 50 feet? Ans. 70.7 ft. + 
7. What is the distance between two opposite corners of a 
square whose area is 900 square feet ? Ans, 42.426 ft. + 


8. What is the distance between two opposite corners of a 
rectangle 15 rods long by 20 rods wide? 

9. What distance will I save by walking directly across, from 
one corner of a plantation a mile square to the opposite corner, 
in stead of following the two sides? Ans. 187.452 rd. 

10. A person lays out two circular plots, one containing 9 times 
~ as much land as the other. How do their diameters compare? 


Cube Root. 


513. Extracting the cube root of a number is resolving 
it into three equal factors; as, 8 = 2 x 2 x 2. 

514, Taking the smallest and the greatest number 
that can be expressed by one figure, by two and three 
figures, let us see how the number of figures they contain 
compares with the number of figures in the cubes :— 

foots, 1 9 10 99 100 999 

Cubes, 1 %29 | 1000 970'299 | 1'000’000 997'002’999 

We find from these examples that, 7f we separate a 
cube into periods of three figures each, commencing at the 
right, there will be as many figures in the cube root as 
there are periods in the cube,—counting the left-hand figure 
or figures, of there are but one or two, as a period. 

515. We derive the method of extracting the cube 
root from the opposite operation of cubing. Cube 25, 
regarding it as composed of 2 tens (20) and 5 units. 


518. What is meant by extracting the cube root of a number ?—514. How can 
we find, from a cube, the number of figures its cube root contains ?—515, Whence 
Ao we derive the method of extracting the cube root? Cube 20 + 5. 


310 EVOLUTION, 


The square of 20 + 5 was found in § 501; we multiply it by 20 + 5. 
Square of 20+5 = 207 + 2 (20 x 5) + 8? 


20 +5 
Multiplying by 20, 20° + 2 (207 x 6) + (20 x 5’) 
Multiplying by 5, (20? x 5) + 2 (20 x 5?) + 5% 


Adding partial prod’s, 20° +3 (20? x 5) + 3 (20 x 5%) + 5° = 25° 
As 5 is a common factor of the last three terms, the cube of 25, as 
just found, may be written as follows :— 


8 times 20? 20? = 8000 
20° + | + 3 times 20 x 5| x 5 8 (207 x 5) = 6000 
+ 5? 3 (20 x 5?) = 1500 


Hence, the cube of a number com- 55 — 125 


posed of tens and units equals the 25° = 15625 


8 times the square of the tens 
cube of the tens + | + 8 times product of tens and units | x the units. 
+ the square of the units 


516. Reverse the process ; find the cube root of 15625. 


According to $514, we separate 15625 into periods of three figures 
each, beginning at the right (15'625), and find that the root will contain 
two figures,—a tens’ and a units’ figure. 


The cube of the tens must be found in the left-hand period 15(000). 
~The greatest number whose cube is contained in 15(000) is 2(0), 

which we place on the right as the tens’ figure of the 
15'625 (2 = root. 2 tens (20) cubed = 8 thousands, which we sub- 
tract from the 15 thousands. Bringing down the remain- 
ing period, we have 7625; which, $515, must equal 


3 times the square of the tens 
+ 3 times the product of the tens and units | x the units. 
+ the square of the units 


“7625 


Hence, to find the units’ figure of the root, we divide 7625 by 3 times 
the square of the tens as a trial divisor. It is contained 6 times; but, 
making allowance for the com- KIQORX (OK 
pletion of the trial divisor, we 15'625 (25 
regard the quotient as 5, and 


write 5 in the root as its units’ ‘Trial div., 20? x 8 = 1200 7625 
ficure. Now, to complete the 20 x 5 x = “4 
divisor, we have to add to 3 Complete divisor, 1585 7625 


times the square of the tens, al- 

ready found, 3 times the product of the tens and units (20 x 5 x 3 = 800), 

and the square of the units (5* = 25),—making 1525. Multiplying thig 

by the units’ figure, and subtracting, we have no remainder. Ans. 25. 
How may the cube just found be written? Hence, what does the cube of : 


number cemposed of tens and units equal ?—516. Reverse the process; extract the 
cube root of 15625, explaining the steps.—d17. Recite the rule. 


CUBE ROOT. | 311 


517. Rurx.—1. Separate the given number into periods 
of three figures each, beginning at the units’ place. 

2. Kind the greatest number whose cube is contained in 
the left-hand period, and place it on the right as the first 
root figure. Subtract its cube from the first period;-and to 
the remainder annex the second period for a dividend. 

3. Lake three times the square of the root already 
Sound ; and, annexing two ciphers, place it on the left 
as a trial divisor. Find how many times the trial di- 
visor ts contained in the dividend (making some allow- 
ance), and annex the quotient to the root already found. 
Complete the trial divisor, by adding to tt 30 times the 


product of the last root figure and the root previously 


Sound, also the square of the last root figure. Multiply 
the divisor, thus completed, by the last root figure, subtract 
the product from the dividend, and bring down the next 
period as before. 

4, Repeat the processes in the last paragraph, tilt the 
periods are exhausted. 

If any trial divisor is not contained in its dividend, place 0 in the root, 
annex two ciphers to the trial divisor, bring down the next period, and 
find how many times it is then contained. 

If, on multiplying a completed divisor by the last root figure, the 
product is greater than the dividend, the last root figure must be dimin- 
ished, and the necessary changes made in completing the divisor. 

Separate a decimal into periods, from the decimal point to the right, 
completing the last period, if necessary, by annexing one or two o™aers, 


To find the cube root of a common fraction, see § 505. 
To prove the operation, cube the root found. 


Ex. 2.—Extract the cube root of 348616. 378872. 
348'616.378'872 (70,38 


Q 
Wx3= 147 843 
1st trial divisor, 14700 5616 
2d trial divisor, 1470000 56163878 
70x8x30 = 6300 
82S 
Complete divisor, 1476309 4498927 
@d trial divisor, 148262700 1187451872 
703 x 8 x 30 = 168720 
§2 = 64 


Complete divisor, 148431454 1187451872 


ta 3 


mag - 3 . ee 
%, ; ant. oe ae { 7 ei F f ra” 
ne ve / ae, b a Pen of; ook : 4 # 


/ a 
é ? 
812 3 EVOLUTION. 


EXAMPLES FOR PRAOTIOE,. 


1, Extract the cube root of 2357947691. Ans. 1331. 

2. What is the cube root of 91125? Of 7256313856? Of 
887420489? Of,10077696 ? Sum of ans. 2926. 
8. Extract the cube root of 42875. Of 125450540216. Of 
848558903294872, Of 117649. | Sum of ans. 75128, 
_ 4, What is the cube root of 18.609625? Of .065450827? Of 
.000000008? Of 1.25992105, carried to five decimal places? Of 
38, to four decimal places ? Sum of ans. 5.57725. 
5. Find the cube root of 243%. Of 2845. Of 174. Of 8%. 
Of 2456. Of 385,09, Of3. Of1014. “Ans. 4%, 14, 2.577 +, &e. 


_ §18. The solid contents of similar bodies are to each 
other as the cubes of their like dimensions. The solid 


» /- contents of two globes whose diameters are 6 in. and 12 


in., are to each other as 6° to 12°, or 216 to 1728. 

519. When the solid contents of a cube are known, ex- 
tract the cube root, to find one of the sides. The answer 
will be in the denomination of linear measure that corre- 
sponds to the denomination of the solid contents. _ A cu- 
bical block whose solid contents are 8 cubdie inches, will 
be 2 (8/8) dinear-inches on each side. 


6. If a ball 3 in. in diameter weighs 8 lb., what will a ball of 
equal density, whose diameter is 4in., weigh ? Ans. 1884 lb. 
7. What is the side of a cube whose solid contents equal those 

of a rectangle, 8 ft. 8 in: long, 3 ft. wide, and 2 ft. 7 in. deep? 
Ans. 47.9848 in. 

8. What is the cide of a cube containing 2197 cu. in.? 

9. There are three balls whose diameters are respectively 38, 4, 
and 5 inches. What is the diameter of a fourth ball, of the same 
density, equal in weight to the three ? Ans, 6in. 

10. If a ball 12in. in diameter weighs 238 1b., what will be 
the diameter of another ball of the same metal, weighing 32 Ib. ? 


518. What principle is laid down respecting the solid contents of similar bodies? 
—t19. How is the side of a cube found from its solid contents ? 


PROGRESSION. 313 


CHAPTER XBOX V IT: 
PROGRESSION. 


520. Progression is a regular increase or decrease in a 
series of numbers. 

521. There are two kinds of f Progression, ethereal 
and Geometrical. 

A series of numbers are said to be in Arithmetical 
Progression, when they increase or decrease by a common 
difference: as, 16, 18, 20, 22; 16, 14, 12, 10. 

A series ee Aiinibers Le said to ie in Geometrical 
Progression, when they increase or decrease by a common 
ratio: as, 16,32, 64, 126; 16, 8, 4, 2. 

522. The numbers forming the series are called Terms, 
The first and the last term are the Extremes, the inter- 
mediate terms the Means. ; 

523. When the terms increase, they form an Ascending 
Series; when they decrease, a Descending Series. 


Avrithmetical Progression. 


524, In Arithmetical Progression, there are five things 
-to be considered: the First Term, the Last Term, the 
Number of Terms, the Common Difference, and the Sum 
of the Series. Three of these being given, the other two 


ean be found. 


To find the relations between these five elements, let us look at the 
series that follow, in which the first term is 13, the common difference 2, 
and the number of terms 5 :—— 

Ascending, 138, 1842, 184242 18384-24242, 13842424242, 
Deseending, 18, 18—2, 183—2—2, 138—2—2—2, 13—2—2—2-2. 

It will be seen that the second term equals the Ist, plus (in the de- 

scending series, minus) once the common difference; the third term 


520. What is Progression ?—521. How many kinds of Progression are there? 
When are numbers said to be in Arithmetical Progression? When, in Geometrical 
Progression? Give examples.—522. What are the numbers forming the series 
called? What are the Extremes? What are the Means ?—523. What is an Ascend- 
ing Series? What is a Descending Series ?—524, How many things are to be\con- 
sidered in Arithmetical Progression? Name them, How many of these must be 
given, to find ths rest? 


14 


814 PROGRESSION, 


equals the 1st, plus (or minus) ¢wice the common difference; the fourth 
term equals the Ist, plus (or minus) ‘Aree times the common difference. 
And, generally, any term equals the first term, increased (or diminished) 
by the common difference taken as many times as the number that rep- 
resents the term, less 1. Hence the following rule :— 


Rue L—TVhe first term, common difference, and num- 
ber of terms being given, to find the last term, multiply 
the common difference by the number of terms less 1, and 
add the product to (or in a descending series subtract tt 
Srom) the first term. 

525, Again, looking at the series, we see that the last term equals 
the first term plus (or minus) the common difference taken as many 
times as there are terms, less 1. Hence the following rules :— 

Rue I.— The extremes and number of terms being 
given, to find the common difference, divide the difference 
of the extremes by the number of terms less 1. 

Rouie U.— The extremes and common difference being 
given, to find the number of terms, divide the difference 
of the extremes by the common difference, and to the quo- 
tient add 1. 


526, To find the average value of the terms of a series, we add the 
extremes (the greatest and the least term), and divide their sum by 2. 
Having thus found the average, if we multiply it by the number of terms, 
we shall have the sum of the series. 

Rutz IV.— The extremes and number of terms being 
given, to find the sum of the series, multiply half the sum 
of the extremes by the number of terms. 

59%, These principles are embodied in the following formulas :—— 

a = first term, Then, J7=at+d x a-— 1). 

= last term, d= =* or ant 

= number of terms, seg BC piece i, 
= common difference, N=-7-+tlor y+. 
= sum of series. pe Se he 


easn 


In solying the examples, ask what is given, and what required, and 
apply the proper rule or formula. 


: Examining the two series that are given, what do we find the second term 
equals? The third? The fourth? What general principle is deduced? Recite 
Rule I.—525. Again, looking at the series, what do we find that the last term equals? 
- Recite Rule Il. Recite Rule I1I.—526. How may we find the average value of the 
terms of a series? How may we find the sum of the series? Recite Rule IY. 


ARITHMETICAL PROGRESSION. 3815 


Ex, 1.—A person made 12 deposits in a bank, increas- 
ing them each time by a common difference. His first 
deposit was $50, and his last $160; what were the inter- 
mediate ones? 


Here we have given the extremes, $50 and $160, and the number of 
terms, 12. The means are required, ‘and to form them we need the com- 
mon difference. Apply Rule II. 


169 — 50 = 110, difference of extremes. 
11011 = 10, common difference. 
$60, $70, $80, $90, &c., intermediate deposits. -Ans. 


Ex. 2.—A falling body moves 16+, ft. during the first 
second of its descent, and 1443 ft. the fifth second. How 
far does it fall in five seconds ? 


Here we have the extremes, 16,4; and 144%, and the number of terms, 
5. The sum of the series is required. Apply Rule IV. 


167; + 1443 = 1608, sum of the extremes. 
1608 - 2 = 802 1%) half the sum of the extremes. 
8075 AGW re = 402,/; ft., whole distance. Avs. 


EXAMPLES FOR PRAOTIOE. 


1. A field of corn containing 50 rows has 20 hills in the first 
row, 23 in the second, and so on in arithmetical progression. 
How many hills in the last row ? Ans. 167 hills. 

2. A person travelling 25 days went 11 miles the first day and 
135 the last, increasing the number each day by a common differ- 
-ence. How far did he travel each of the intervening days, and 
how far in all? Last ans. 1825 miles, 

3. A note is paid in annual instalments, each less than the 
previous one by $3. The first payment being $49, and the last 
$7, how many instalments were there? Ans. 15. 

4. A man has 7 sons, whose. ages are in arithmetical progres- 
sion. The eldest being 23, and the youngest 5, what is the differ- 
ence in-age between the youngest and his next elder brother ? 

5. Bought 100 yd. of cloth. The first yard cost £1 15s. 6d., 
and each of the others 4d. less than the preceding one. What 
did the last yard cost, and what the whole? First ans. 2s, 6d. 

6. What is the 20th term of the series, 8, 15, 22, &c. ? 


316 | PROGRESSION. 


Geometrical Progression. 


528. In Geometrical Progression, there are five things 
to be considered: the First Term, the Last Term, the 
Number of Terms, the Ratio, and the Sum of the Series. 
Three of these being given, the other two can be found. 

529, Look at the following series,~in which the first term is 6, the 
ratio 2, and the number of terms 5 :— 

6, 86 x DSB IK FD ADs ONG ID, ED KD IG KK SE Die Bs 
Or, 6, 6 x 2, 6. 27, Gotz. Ox, 

Tt will be seen that each term consists of the first term, 6, multiplied 
by the ratio, 2, raised to a power whose index is 1 less than the number 
of the term. Hence the following rule :— 

Rute I. The first term, ratio, and number of terms 
being given, to find the last term, multiply the first term 
by that power of the ratio whose index is 1 less than the. 


number of terms. 


530. Suppose the sum of the series 6, 18, 54, 162, 486, is required, 
Multiplying each term by 3 (the ratio), we form a second series whose 
sum is 3 times as great. Then, subtracting the Ist series from the 
2d, we have a result twice as great as the sum of the 1st series. 

18, 54, 162, 486, 1458 = 8 times. 
6, 18, 54, 162, 486, = once. 
1458—6 = twice. 
Cancelling the intermediate terms, we have 1458 — 6 for the result, 
which, being twice as great as the sum of the 1st series, we divide by 2. 
Now 1458 is the last term multiplied by the ratio (486 x 8); and 2, by 
which we divide, is the difference between the ratio and 1(8 — 1). Hence, 


Rovrzr I.— The extremes and the ratio being given, to 
jind the sum of the series, multiply the last term by the 
ratio, find the difference between this product and the first 
term, and divide it by the difference between the ratio 


and 1. “4 
FORMULAS :— geet pi a 
7” = ratio g — UEDE One Op tee 
mea 1-?r 


528. How many things are to be considered in Geometrical Progression? Name 
them. How does the ratio compare with 1 in an ascending series? How, in a de- 
scending series ?—529. Looking at the given geometrical series, of what will it be 
seen that each term consists? Recite Rule L—530. Go through the reasoning by 
which the rule for the sum of the series is arrived at. Recite Rule IL 


GEOMETRICAL PROGRESSION. se 


al 


the last term is infinitely small, and may be regarded as 0. 
Ex. 1.—What is the amount of $250, for 6 vse at. 
6%, compound interest ? 


531. In a descending infinite series, as 1, 4, 4, 3, &e., 


The principal is the first term of a geometrical series. The amount 
of $1, for 1 year, at 6 4, is the ratio. 6 (years) + 1 (the principal being 
the first term) is the number of terms. The amount required is the last 
term. Apply Rule IL, § 529. 

1.06° = 1.418519112256 
1.418519112256 x 250 = $354.629 Ans. 


Ex. 2.—What is the sum of a series ef 8 terms, com- 
mencing 200, 50, 124, &c.? 

Here the first term, the ratio (50 + 200 = 4), and the number of 
terms are given, and the sum of the series is required. We first apply 


tule I., $529, to find the last term; and then Rule II., § 530, to find 
the sum. 


1 AY, eg 1 S 
) — T6354 TUsSE x 200 = —_ goes las t ter Im. 
2 x 1 Bios 


EXAMPLES FOR PRACTIOE. 


1. A person goes 24 miles the first day, 5 the second, and so 
on in geometrical progression. If he travels thus for 8 days, how 
far will he go the last day? How far in all? Last ans. 6374 mi. 

2. B invested $1000 so that it would double itself every four 
years. What did his capital amount to at the end of the twelfth 
year? At the end of the twentieth year? 

3. What is the amount of $800, for 5 years, at 7%, compound 
interest ? Ans. $1122.04. 

4, If ten stones are laid in a line, the first 3 ft. from a basket, 
the second. 9, the third 27, and so on in progression, how far must 
a person starting from the ‘basket walk, to pick them up singly 


and place them in the basket ? Ans. 33434 mi. 
5, First term, 100; ratio, +; number of terms, 9. Ree 
the sum of the series. Ang. 199 22. 


6. Find the sum of the infinite series 1, 3, 4, 4, &c. ($531). 


7. Find the sum of the infinite series 1, 4, 4, &e. Ans. 14. 
I 


318 MENSURATION, 


CYLAP'T HAR Xx Xe DT 
MENSURATION. 


532, Mensuration is that branch which gives rules for 
finding the length of lines, the areas of surfaces, and the 
solidity of bodies. These rules are derived from Geometry. 


Several rules of Mensuration have been already given; as, those re- 
lating to the sides of right-angled triangles, $512. Some of the others 
that are most important are given below. 


533. ParaLLELoGrams.—A A 
Parallelogram is a four-sided Po abek HUES LT 
figure that has its eppeate 
sides equal and parallel. 
square and a rectangle are ee eee 
The Base of a parallelogram is the side on which it 


stands. Its Altitude is the perpendicular distance from 
its base to the opposite side; as, A B in the figures. 


Rutz.— To find the area of a parallelogram, multiply 
the base by the altitude.. 


1. How many square feet of surface will be covered by 12 


boards 18 ft. long and 18 in. wide? Ans. 324 sq. ft. 
2. Find the cost of a piece of land 40 ch. 151. square, at $30 
an acre. Ans. $4836.0675. 


8. What is the difference between the areas of two parallelo- 
grams, the one 80ft. long and having an altitude of 20ft., the © 
other having a length of 30ft. and an altitude of 25 ft. ? 


534, TrranciEes.—The Altitude of a ¢ C 
Triangle is a perpendicular drawn from 
one of its angles to the base, or the base 
produced ; as, C D. 

Rvutz— Zo find the arca of a triangle, multiply is 
base by half its altitude. 


532. What is Mensuration ?—533. What is a Parallelogram ? What is its Base? 
What is its Altitude? Recite the rule for finding the grea of a parallelogram.— 
534. What is the Altitude of a Triangle? Recite the rules for finding the area of a 
triangle. 


re us D 


MENSURATION. old 


Or, wlien the three sides are given, from half their sum 
subtract cach side separately, multiply together the three 
remainders and the half swm, and extract the square root 
of their product. 


4, What is the arca of a triangle whose base is 12 feet and its 


altitude 8 yards? Ans. 54 sq. ft. 
5. What is the area of a triangle whose sides are respectively 
7, 11, and 12 feet? Ans. 37.94 sq. ft.+ 


6. In a triangular field whose sides are 18, 80, and 82 feet, 
how many square yards? 


535. Crrctes.—The Circumference, Diameter, and 
Radius of a Circle are defined on page 157. 

Rures.—I. To find the circumference of a circle, mul- 
tiply the diameter by 3.14159. 

Il. Zo jind the diameter, multiply the circum ference 
by .3183. 

Ill. Zo jind the area, multiply + ihe cir OT by 
the diameter. 

Or, multiply the square of the circumference by .07958, 

Or, muitiply the square of the diameter by .7854, 


7. The diameter of the earth being 7926 miles, what is its cir- 


cumference ? _ Ans. 24900.24234 mi. 
8. Over what distance will a wheel 4 ft. 9 inches in diameter 
pass, in making four revolutions ? . Ans. 59.69021 ft. 


9. If the tire of a wheel is 14.8235 ft. in circumference, what 
is its diameter ? 

10. What is the area of a circular plot requiring 40 rods of 
hedge to enclose it ? 

11. If I describe a circle with a rope 40 ft. long, fixed at one 


end, what will be its area? Ans, 5026.56 sq. ft. 
12. A circle contains 415.4766 sq. inches, what is the square 


of its diameter? What is-its diameter ? Last ans. 23 in. 


535. What is the Circumference of a circle? The Diameter? The Radiuww? Re 
cite the rule for finding the circumference, The diameter. The area, 


320 MENSURATION. - 


536, Cyrinpers.—A Cylinder is a body of uniform 
diameter, bounded by a curved surface, and two 
equal and parallel circles, either of which may 
be regarded as its base. 

The Altitude of a cylinder is the perpendic- 
ular distance between its bases. 

Roies.—I. 70 find the surface of a cylinder, 
multiply the circumference of the base by the altitude, and 
to the product add twice the area of the base. 

Il. To find the solidity of a cylinder, multiply the 
area of the base by the aititude. 


The base being a circle, its area-‘may be found by Rule III., § 585 


18. How many square feet in the surface of a stove-pipe 20 
feet long and 5 inches in diameter ? Ans. 26.179 sq. ft. + 
14. Ilew many gallons (wine) will a cylindrical cistern hold, 
that is 15 ft. deep and 4 ft. across? Ans. 1410.048 gal. 
15. A cylindrical piece of timber is 24 feet long and 18 inches 
across; what will it cost, at 20c. a cubic foot? Ans. $8.48. 


5387, Spueres.—A Sphere is a body bounded es a 
curved surface, every point of which is equally 
distant from a point within, called the centre. 

Ruies.—I. To find the surface of a sphere, 
multiply the square of the diameter by 8.14159. 

Il. Zo jind the solidity of a sphere, multiply the cube 
of the diameter by .5236, 


16. Required the surface and solidity of a sphere 30 inches in 
diameter. Ans. 19 sq. ft. 91.481 sq. in.; 8 cu. ft. 313.2 cu. in. 

17. The diameter of the earth is 7926 miles; if it were a per- 
fect sphere, how many square miles would its roe contain ? 

18. Required the solidity of a sphere 2 yd. in diameter. How 
many square yards in its surface ? 


536. What is a Cylinder? What is the Altitude of a cylinder? Reeite the rule 
for finding the surface ofa cylinder. For finding the solidity of a cylinder.—5387. 
What isa Sphere? Give the rule for finding the surface of a sphere, For finding 
the solidity of a sphere, 


ANNUITIES, 321 


CHAPTER XXXIX. 


ANNUITIES. 


538, An Annuity is a sum payable yearly. 

A Certain Annuity is one payable for a definite num- 
ber of years. 

A Life Annuity is one payable yearly during the life 
of a person or persons. 

A Perpetuity is an annuity payable yearly forever. 

An Annuity in Reversion is one that is to commence 
at some future time. 

An Annuity Forborne, or in Arrears, is one that re- 
mains unpaid after it is due. 

539. The Amount of an Annuity Forborne is the sum 
of the amounts of the several payments due, for the time 
they have remained unpaid. 


540. The Final Value of a Certain Annuity is the 
sum of the amounts of the several payments, computed 
from their date to the expiration of the given time. 

The Present Value of a Certain Annuity is such a 
sum as, put at interest for the given time and rate, would 
amount to its Final Value. | 


541. To find the amount of an annuity forborne, or 
the jinaé value of a certain annuity 

Hither simple or compound interest may be allowed. 
The latter is more usual, and a Table is then used in the 
computation with great advantage. 


542. Rutz I.—If simple interest only is allowed, Mul- 
tiply the annuity by the number of payments due, and to 
the product add the interest of the annuity for a term 


538, - Whatis an Annuity? What isa Certain Annuity? A Life Annuity? . 
Perpetuity? An Annuity in Reversion? An Annuity Forborne ?—539. What is 
meant by the Amount of an Annuity Forborne ?—540. What-is meant by the Final 
Value of a Certain Annuity? By its Present Value ?—541. In computing the amount 
ef an Annuity Forborne, what kind of interest is usually allowed ?—542. Recite the 
tule for finding the amount of an annuity that draws simple interest. Explain Ex. 1. 


322 ANNUITIES. 
equal to the sum of all the periods during which successive 
payments are due. 

Ex. 1.—What is the amount of an annuity of $750, in 
arrears 5 years, at 7%, simple interest ? 


Five payments are due: $750x5 = ; $3750 
Int. is due on the 1st payment for 4 years. 
oe (7 ee 2 9 “cc (79 od (7 (79 8 years. 
66 7 See 93 (7 (79 8d ae (79 9, years. 
66 (7 Se 93 “6 (79 4th (a9 (79 1 year. 

Interest on $750, at 7%, for 10 years, . 525 


Amount due, at simple interest, $4275 


543. Rutz Il.—If compound interest is allowed, Jful- 
tiply the amount of $1 for the given time and rate, in the 
following Table, by the annuity. 

TABLE, 
showing the amount of an annuity of $1 or £1, up to 21 years, at 


Yr. 


CO OTS OU to DH 


3, 4, 5, 6, and 7%, compound interest. 


8 per ct. 


1.000000 


2.030000 
8.090900 
4.183627 
5.809136 
6.468410 
7.662462 
8.892336 
10.159106 
11.463879 
12.807796 
14.192030 
15.617790 
17.086324 
18.598914 
20.156881 
1.761588 
23.414.435 
25.116868 


—26.8703874 


28.676486 


4 per ct. 


1.000000 


2.040000 
3.121600 
4.246464 
5.416323 
6.632975 
7.898294 
9.214226 
10.582795 
12.006107 
13.486351 
15.025805 
16.626838 
18.291911 
20.023588 
21.824531 
23.697512 
25.645413 
27.671229 
29.7'78079 
31.969202 


5 per ct. 


1.000000 


2.050000 
3.152500 
4.310125 


5.525631 


6.801913 

8.142008 

9.549109 
11.026564 
12.577898 
14.206787 
15.917127 
17.712983 
19.598682 
21.578564 
23.657492 
25.84.0366 
28.132385 
30.539004 
33.065954 
85.719252 


6 per ct. 


1.000000 


2.060000 
3.183600 
4.374616 
5.637093 
6.975319 
8.393838 
9.897468 
11.491316 
13.180795 
14.971643 
16.869941 
18.882138 
21.015066 
23.275970 
25.672528 
28.212880 
30.9056538 
33.759992 
36.785591 
39.992727 


7 per ct. 


1.000000 


2.070000 | 

8.214900 | 

4.439943 

5.750789 

7.153291 

8.654021 
10.259803 
11.977989 
13.816448 
15.783599 
17.888451 
20.140643 
29.550488 
25.12.9022 
27888054 
30,840217 
33.999033 
37378965 
40.995492 
44,865177 | 


y 


543. Recite the rule for finding the amount of an annuity that draws compound 


interest. 


Explain Ex. 2. 


ANNUITIES. 323 


Ex. 2.—What is the final value of an annuity of $250, 
for 20 years, at 6%? 


Amount of $1, for 20 years, at 6 4, in Table, $36.785591. 
$36.785591 x 250 = $9196.39775 Ans. 


544, EXAMPLES FOR PRAOTICE. 


1. Required the amount of an annuity of $1000, in arrears for 
6 years, at 7 4%, simple interest. Ans. $7050. 

2. What is due to a clerk whose yearly salary of $400 has re- 
mained unpaid 5 years, allowing 6 %, simple interest ? 

3. A tenant holding property for which he is to pay $275 at 
the end of every year, pays no rent for 10 years. How much does - 
he then owe, at 5%, simple interest? How much, at 64%, com- 
pound interest ? Last ans. $3624.72. 

4, A gentleman, on the birth of a son, and on each subsequent 
birthday, deposits $25 to his credit in a savings bank that allows 
5 % compound interest. How much will stand to the son’s credit 
when he reaches the age of 20? 

5. A lawyer collects for A the amount of an annuity of $100 
forborne 7 years, with simple interest, at 7%; and for B the 
amount of an annuity of $325, in-arrears 3 years, with compound 
interest at 6%. He charges them both 10% on the amount collect-_ 
ed; how much more does he receive from Bthan A? Ans, $18.77. 


545. To find the present value of an annuity. 


546, The present value of a perpetuity is evidently a 
sum,the annual interest of which will be the given perpe- 


tuity. It may be found by § 352. 


What is the present value of a perpetui 
6%? That is, what sum, at 6 Z, will yield 
06 = $1000 Ans. 

547, Ruty.—To find the present value of a certain 
annuity, Multiply the present value of $1, for the given 
time and rate, in the following Table, by the annuity. 


ty of $60, money being worth 
%60 a year forever? $60+ 


546, What is the present value of a perpetuity? Give an exam ple.—547, Recite 
the rule for finding the value of a certain annuity. Explain the example, 


324 ANNUITIES. 


TABLE, 


showing the present value of an annuity of $1 or £1, up to 21 


Kd : 


Saracen | F 


years, at 3, 4, 5, 6, and 7%, compound interest. 


3 per ct. 


0.970874 | 


1.913470 
2.828611 
3.717098 
4.579707 
5.417191 
6.230283 
7.019692 
7.786109 
8.530203 
9.252624 
9.954004 
10.634955 
11.296073 
11.937935 
12.561102 
13.166118 
13.753513 
14.323799 
14.877475 
15.415024 


4 per ct. 


0.961538 


1.886095 
9.775091 
3.629895 
4.451822 
5.242137 
6.002055 
6.732745 
7.435332 
8.110896 
8.760477 
9.385074 
9.985648 
10.563123 
11.118387 
11.652296 
12.165669 
12.659297 
13.133939 
13.590326 
14.029160 


5 per ct. 


0.952381 
1.859410 
2.723248 
8.545951 
4.329477 
5.075692 
5.786373 
6.463213 
7.107822 
7.721735 
8.306414 
8.868252 
9.393573 
9.898641 
10.379658 
10.837770 
11.274066 
11.689587 
12.0853821 
12.462210 
12.821153 


6 per ct. 


0.943396 


1.883393 
2.673012 
8.465106 
4.212364 
4.917324 
5.582381 
6.209744 
6.801692 
7.360087 
7.886875 
8.383844 
8.852683 
9.294984 
9.712249 
10.105895 
10.477260 
10.827608 


11.158116 . 


11.469421 
11.764077 


q atee Ch, 


0. ~ 0.934579 
1.808017 
2.624314 
3.887209 
4.100195 
4.766537 
5.389286 
5.971295 
6.515228 
7.023577 
7.498669 
7.942671 
8.857635 
8.745452 
9.107898 
9.446632 
9.763206 
10.059070 
10.335578 
10.593997 
10.835527 


= 


Ex. How much should a person receive, cash down, 
for an annuity of $150, to run 15 years, at 5%? 


Present value of $1, for 15 yr., at 54%, in Table, $10.379658. 
$10. 379658 x 150 = $1556.9487 ‘Ans. 


EXAMPLES FOR PRAOTICE. 


1. Find the present value of an annuity of $950, to run nine 
Ans. $6752.48, 


years, at 5%. 


2. What sum, cash down, should a widow reccive for an an- 
nuity of $500, to run 20 yr., computing at 7 4? 
3. When permanent investments command 64%, how much 
must a person present to a college, to enable it to award a prize 
of $150 yearly forever ? - 


Ans. $2500. 


9 


ANNUITIES. 325 


4, A gentleman wishes to secure an annuity of $300 to each 
of his four children for cighteen years. What must he pay to do 
so, computing at 5 %? Ans. $14027.50. 

5. How much must a lady invest, at 4%, to secure a perpetu- 
ity of £50 to the poor of her native town? 3 

6. Which is worth most, $2500 down, $2850 payable in 2 
years, or an annuity of $220, to run 21 years, computing at 6 4? 

7. What is the difference between the amount of $200 at com- 
pound interest for 5 years, at 7%, and the amount of an annuity 


of $200, to run 5 years, computed at 7 4 compound interest ? 
Ans. $869.6374. 


CHAPTER XL. 
THE METRIC SYSTEM. 


548. By the Metric System is meant a decimal system 
of weights and measures, used to the exclusion of. all 
others in France, Belgium, Spain, and Portugal, and to 
some extent in other countries of Europe. It has been 
legalized in Great Britain, Mexico, and many of the 
South American states; and in 1866 an Act was passed 
by the Congress of the United States, SUELGERE 1) its 

use. 


The advantages that would result from a universal adoption of this 
system are obvious. First, there would be no necessity for converting 
the denominations of one country into those of another, for all would 
have the same; and, secondly, the system being decimal throughout, all 
operations in Reduction and Compound Numbers would be performed 
with the same ease that they now are in Federal Money. 


549, The unit of length is the Mrrre, from eg the 
Metric System derives its name. The metre is zy5qy000 


548. What is meant by the Metric System? Where has it. been legalized? 
What Act was passed in 1866? What advantages would result from a universal 
adoption of the Metric System ?—549. According to the Metric System, what is the 
unit of length? To what is it equal? 


326 THE METRIC SYSTEM, 


of the circumference of the earth measured over the poles, 
and is equal to 39.37 inches. 

950. The unit of surface is the arz (pronounced air), 
which is a square whose side is 10 metres, and equals 
119.6 square yards. 

551. The unit of capacity is the trrrz (pronounced 
le'tur), which is a cube whose edge is +1, of a metre, and 
equals .908 of a quart dry measure, or 1.0567 quarts 
liquid measure. 

552, The unit of weight is the Gram, which is the 
weight, in a vacuum, of a cube of pure water whose edge 
is z}y Of a metre. The gram equals 15.432 grains. 

553. From these principal denominations are formed 
others 5, z}a, and sop as great, denoted by the prefixes 
DECI, (pronounced des‘e), cENTI, and MILLI;~ also, other 
denominations 10, 100, 1000, and 10000 times as great, 
denoted by the prefixes DECA, HECTO, KILO, and MYRIA. 


Measures oF Lenarn. 


10 millimetres make 1 cen’timetre = .3937 inch. 

10 centimetres “ 1 dec'imetre _=38.937 inches. 
10 decimetres ‘“ 1 ME’TRE = 39.87 inches. 
10 metres ¢ 1dec’ametre = 82 ft. 9.7 in. 


10 decametres ‘“ 1 hec’tometre = 328 ft. 1 in. 
“10 hectometres ‘“* 1 kil’ometre = 8280 ft. 10 in. 
10 kilometres ‘ 1myr'iametre = 6.21387 miles, 


MEASURES OF SURFACE. 


The cen’tiare is 1 square metre, or 1550 square inches. 
100 centiares make 1 ARE == 119-6 80. vd. 
100 ares “1 hee’tare = 2.471 acres, 
No other denominations are used. 


550. What is the unit of surface? To what is it equal ?—551. What is the unit 
of capacity ? To what is it equal ?—552. What is the unit of weight? To what is 
it equal?—553. From these principal denominations, what others are formed, and 
how? What is the unit used in the measurement of wood, and to what is it equal? 


THE METRIC SYSTEM. 324 


MEASURES OF CAPACITY. 


Dry Measure. Liquid Measure. 
.6102 cu. in. = .838 fluid 2. 


10 millilitres, 1 cen’tilitre = 33! 
6.1022 cu. in. = .845 gill. 


10 centilitres, 1 dec’ilitre 


Holl Il 


10 decilitres, 1 Li'TRE 908 gt. = 1.0567 qt. 
10 litres, 1 dec’alitre = 1.185 pk. = 2.6417 gall. 
10 decalitres, 1 hee’tolitre = 2.8375 bu. = 26.417 gall. 


10 hectolitres, 1 kil’olitre =1.808 cu. yd. = 264.17 gall, 


The kilolitre (1 cubic metre), when applied to the measure- 
ment of wood, is known as the STERE, Which equals .2759 cord. 
10 steres make 1 dec’astere = 2.759 cords. 


WEIGIITS. 


10 mil'ligrams, 1 cen’tigram = .15482 gr, 


10 centigrams, 1 dec'igram = 1.5432 er. 
10 decigrams, 1 GRAM = 15.482 er. 
10 grams, 1 dec’agram = .8527 oz. av: 


10 decagrams, 1 hec'togram = 3.5274 oz. av. = 3.214 oz. Troy. 
10 hectograms, 1 kil’ogram = 2.2046 lb. av. = 2.679 lb. Troy. 
10 kilograms, 1 myr‘iagram = 22.046 lb. av. = 26.79 Ib. Troy. 
10 myriagrams, 1-quintal = 220.46 Ib. av. = 267.9 Ib. Troy. 
10 quintals, ltonneau =1:1023 ton = 2679 lb. Troy. 


554, Exmrcise oN Tun Metric System. 


1. Write 2 hectograms 5 grams 8 decigrams 7.centigrams as 
grams and the decimal of a gram. Ans. 205.37 grams. 
2. Write 6 kilolitres 7 decalitres 8 litres 8 decilitres 4 milli‘ 
litres as litres and the decimal of a litre. 
3. Write as metres 2 myriametres 3 hectometres 4 decametres 
5 metres 6 centimetres 9 millimetres. 
4, Write as ares 2 hectares 9 centiares. 
. How many milligrams in a tonneau? In a gram? 
. Reduce 3 hectares 5 ares to centiares. 
. How many litres in 74 decalitres? In 2 kilolitres ? 
. How many decigrams in a decagram? In a quintal? 
. Which is greater, 1 rod or 5 metres? 1 decimetre or 4 


may co Ol 


328 TME METRIC SYSTEM. 


inches? 1 litre or 1 quart? 2 hectares or 5 acres? 1 ounce avoir. 
dupois or 80 grams? 4 steres or 1 cord? About how many myr 
iagrams equal 1 cwt. ? 

10. What is the area of a rectangular floor 5 metrés wide and 
7 metres long? | Ans. 35 centiares. 

11. What is the area of a square 20 metres on each side? 

12. How much land is there in a rectangular lot 9 metres by 2 
decametres ? Ans. 1.8 ares. 

13. How much wood will a crib hold that is 1 metre in length, 
width, and height ? Ans. 1 stere. 

14. How much wood in a pile 5 metres long, 1 metre wide, 
and 2 metres high ? Ans. 1 decastere. 

15. How many decasteres in a pile of wood 10 metres long, 1 
metre wide, and 3 metres high ? 

16. A druggist, having a hectogram of calomel, puts up from it 
20 powders of 1 gram 2 decigrams each; how much calomel has 
he left ? 

17. What is the cost of 7 kilograms 2 hectograms of butter, at 
75 cents a kilogram? ($.75 x 7.2) Ans. $5.40. 

18. What is the cost of 11 metres 7 decimetres of silk, at $2.80 
a metre ? 

19. A person gives $10500 for 8 hectares of land, and Jays it 
out in lots of 24 ares each. What must he sell it for per lot, to 
make 20 %? Ans. $39,375. 

20. A grocer buys a quintal of butter for $88, and retails it for 
$1.10 a kilogram. Tow much does he make on the whole, and 
what per cent. ? Last ans. 25 %. 

21. How many hectolitres of potatoes, at 18 cents per deca- 
litre, must a farmer give for 3 kilograms 4 hectograms of tea, at 
$2.40 per kilogram ? Ans. 4,8; hectolitres. 

22. How many kilometres of wire fence will be ‘required, te 
eons a square field 350 metres on each side ? 

3. What will be the profit on 14 hectolitres of beer, bought 
for $5 per hectolitre and retailed at 8 ¢. per litre ? 

24. From 12 decasteres of wood were sold 11 steres. Re- 
quired the value of what remained, at $3 per stere. Ans, $3277, 


MISCELLANEOUS QUESTIONS. _ 329 


MISCELLANEOUS QuEstions.—In Percentage, what three things are to 
be considered? Give the three formulas that apply to the percentage, 
rate, and base. In Profit and Loss, what correspond to the percentage, 
rate, and base? Give three formulas, then, corresponding to those in 
Percentage, that will apply to Profit and Loss. In Interest, what corre- 
spond to the percentage, rate, and base? Make three formulas, then, 
corresponding to those in Percentage, that will apply to Interest. In 
Commission, what correspond to the percentage, rate, and base? Make 
three formulas, then, that will apply to Commission. In Insurance, what 
correspond to the percentage, rate, and base? Make three formulas, 
then, that will apply to Insurance. Which kind of duties involve the 
principles of Percentage? Make three formulas that will apply to ad va- 
lorem Duties. 

In Percentage, how do you find the base, when the rate and the sum 
or difference of the percentage and base are given? To what in Percent- 
age does the selling price of an article sold at a profit correspond? To 
what does the selling price of an article sold at a loss correspond? Give 
the rule, then, for finding the cost, when the selling price and the rate of 
profit or loss are known. 

In Duodecimals, what different values may the prime have? What 
denomination of duodecimals is equivalent to the inch? To the square 
inch? To tke cubic inch? To what is the index of a product in duo- 
decimals equal ? 

What is the shortest method of finding the interest of any sum for 60 
days, at 6%? For 80 days? For3 days? For 63 days? For 90 days? 
For 93 days? For 83 days? How can we find in how many years any 
principal will double itself at a given rate? At what rate will any prin- 
cipal double itself in a given number of years ? 

What is the difference between a promissory note that says nothing 
about interest, one that has the words ‘‘ with interest,’’ and one that says 
“‘ with interest annually’? ? What is the difference between true discount 
and bank discount ? 

How many terms enter into a simple ratio? How many into a simple 
proportion? How many terms in a simple proportion must be given, to 
find what is not given? If the term not given is an extreme, how do we 
find it? How, if itisa mean? What is the difference between a simple 
and a compound proportion? Why do we make that which is of the 
same kind as the answer the third term?. By what other process may ex- 
amples in proportion be solved? What is the ratio of the diameter of a 
circle to its circumference? Of the diameter to the radius? Of the cir. 


cumference to the radius ? 


820 MISCELLANEOUS EXAMPLES, 


555. MIscELLANEOUS EXAMPLES. 


1. If C and D retired at the same hour daily, but C rose at 4 
before 6 and D at half past 7, how much more working time had 
C than D in the years 1864 and 1865? Ans. 12794 hr. 

2. How many acres, roods, &c., in a rectangular field 12 ch. 
34]. long and 10 ch. 85 1. wide? Ans. 13 A. 1 R. 22.224 sq. rd. 

8. If 1 gal. 1 qt. 2 gi. of liquid passes through a filter in 1 hour, 
how much will pass through in 4 hr. 19 min. 24 sec. ? 

Ans. 5 gal. 2 qt. 1 pt. 1.58 gi. 

4, How many square feet of glass in 12 windows, each having 
12 panes, and each pane being 1 ft. 3’ by 11’? Ans. 165 sq. ft. 

5. A grocer sold 104% of his stock of sugar, and then 104 of 
what was left. 60 cwt. 75 lb. remained; what was his original 


stock ? Ans. 75 cwt. 
6. If I divide 4 of a section of land into 13 equal parts, how 
many acres, &c., in each part ? Ans. 16 A. 1 R. 2528 P. 


7. Sold, Mobile, Feb. 1, 1866, 50 bales of cotton, averaging 
426 lb. to the bale, at 45c. a pound. May 15, 1866, received the 
money, with legal interest from the date of sale. How much did 
I receive ? Ans. $9806.52. 

8. Find the amount of £1600 14s. 8d., at 4%, for 18 days. 

9. When it is 10 minutes past 6 o’clock at Chicago, it is 22 
min. 43 sec. past 6 at Cincinnati. What is the difference of longi- 
tude between these cities? Ans. 3° 10’ 45”. 

10. A New York merchant, having £350 to pay in London, 
buys a draft for that amount with gold at 150, exchange standing 
at 109. He might in stead have remitted 5-20’s, then selling at 
104 in N. Y. and worth 64 in London. Would he have gained 
or lost by so doing, and how much? Ans. Gained $15.55. 

11. A rectangular piece of land containing half an acre is five 
times as long as it is broad. Required its length and breath. 

Ans. Length, 20 rd.; breadth, 4 rd. 

12. In a mixture of wine and cider, } of the whole + 25 gal- 
lons was wine, and 4 of the whole —5 gallons cider. How many 
gallons were there of each? Ans. 85 gal. wine, 35 gal. cider. 


MISCELLANEOUS EXAMPLES. sol. 


13. Divide $2000 into shares that shall be to each other as 
8, 7, 6, and 4. Ans. $744.1825, $651.1642, $558.1324, $46. 51 y's. 
14. How many rods of hedge will be required to, enclose a cir- 
cular plot containing lacre? To enclose a square plot containing 
an acre? To enclose an acre in the form of a right-angled trian- 
gle whose altitude is twice its base ? 

Ans. 44.83 rd. +; 50.596rd. +; 66.231 rd. + 
15. What will be the length of a diagonal from a lower corner 

to the opposite tees corner of a cubical vat 9 feet on each side ? 
Ans. 15.58 ft. + 
16. There are two globes, one having a diameter of 10in., the 
other a circumference of 37.69908 in. How many more square in- 
ches in the surface of one than in that of the other? How many more 
cubic inches in one than inthe other? First ans. 138.22996 sq. in. 
17. A person spent £100 for some geese, sheep, and cows, pay- 
ing for each goose 1s., for each sheep £1, and for each cow £5. 
How many did he purchase of each kind, so as to have 100 in 
all ? Ans. 80 geese, 1 sheep, 19 cows. 
18. A hare is 50 leaps before a hound, and takes 4 leaps to the 
hound’s 8, but 2 leaps of the hound are equal to 3 of the hare’s. 
How many leaps must the poe take, before he catches the 
hare ? Ans. 800 leaps. 
19. A general, wishing to draw up his men in a square, found 
on the first trial that he had 89 men over. The second time, hav- 
ing placed one more man in rank, he needed 50 to complete the 
square. How many men had he? Ans. 1975 men. 
20. A, B, and ©, start from the same point, and travel in the 
same direction, round.an island 20 miles in circumference. A 
goes 3 miles an hour, B 7, and C11. In what time will they all 
be together ? Ans. At the end of 5 hours. 
21. From a cask containing 10 gallons of wine, a servant drew ~ 
off 1-gallon each day, for five days, each time supplying the de- 
ficiency by adding a gallon of water. Afterwards, fearing detec- 
tion, he again drew off a gallon a day for fiye days, adding each 
time a gallon of wine. How many gallons of water still remained 
in the cask ? Ans. 2.418115599 gal, 


832 MISCELLANEOUS EXAMPLES, 


22. If I purchase $1200 worth of goods, 4 on 8 months’ credit, 
+ on 6 months, and + on 9 months, what amount in cash would 
pay the bill, money being worth 7%? Ans. $1159.64. 
23. In the above example, what would be the equated time 
for paying the whole amount, $1200, at once? 
24. How many times will the second-hand of a watch go round 
its circle, in 12 wk. 2 hr. 15 min. ? Ans. 121095 times. 
25. Find the sum of the infinite series 1, 3, ,,:&c. Ans. 1t. 
26. A and B had the same income. A saved +} of his; but 
B, spending $120 a year more than A, at the end of 10 years was 
$200 in debt. What was the income ? Ans. $500. 
94%. A father gave his five sons $1000, to divide in such a way 
that each should have $20 more than his next: younger brother. 
What was the share of the youngest ? Ans. $160. 
28. A and B, starting from opposite points of a fish-pond 536 
feet in circumference, begin to walk around it at the same time, 
in the same direction. A goes 62 yards a minute, B 68 yards. In 
what time will B overtake A, and how far will A have walked? 
Last ans. 923% yd. 
29. A, B, and OC, commence trade with $3053.25, and gain 
$610.65. The sum of A’s and B’s capital is to the sum of B’s and 
C’s as5 to 7; and C’s capital diminished by B’s, is to C’s increased 
by B’s, as 1 to 7. What is each one’s share of the gain ? 
Ans. A’s, $185.70; B's, $208.55; C's, $271.40. 
30. A man left $1000 to be divided between his two sons, one 
14 years old, the other 18, in such a proportion that the share of 
each, being put at interest at 6%, might amount to the same sum 
when they reached the age of 21. How much did each receive? 
Ans. Elder, $546.15; younger, $453.85. 
31. If $100 is divided between D, E, and F, so that E may 
have $3 more than D, and F $4 more than E, how much will 
each have ? : Ans. D, $30; E, $33; F, $87. 
32. P and Q have equal incomes. P contracts an annual debt 
amounting to + of his; Q lives on 4 of his, and at the end of 10 
years lends P enough to pay off his debt, and has $320 left. What 
is the income ? Ans. $560, 


MISCELLANEOUS EXAMPLES. 333 


33. What is the final value, and what the present value, of an 
annuity of £80, to run 6 years, computing at 5 4, compound in- 
terest? Last ans. £406 1s. 14d. 

34. What is the difference between 5% and .5 4 of £177? 

35. A buys of D 840 barrels of flour, at $10.50 a barrel, on 
credit. Three days afterward he fails, having $52070 net assets, 
and liabilities to the amount of $254000. How much will D re- 


ceive for his debt ? Ans. $1808.10. 
36. What will it cost to gild a ball 14 yd. in diameter, at $1.75 
a square foot? Ans. $111.83. 


37. A merchant buys, on 4 months’ credit, 8 tons of lead, at 
$10.25 per 100 Ib.; 750 Ib. horse-shoe iron, at $150 a ton; 44 
ewt. of spelter, at 11 ¢. a lb.; and 250 lb. of copper, at 29 c. a Ib. 
What sum, cash down, will pay the bill, computing at 6 4? 

88. With how long a rope must a goat be fastened, that it may 
feed on 345 of arood of land? Ans. 18 ft. 7.38 in. 

89. A person imported 5 casks of wine, averaging 43 gal. each, 
which cost him $1.50 a gallon in gold. The duty was $1 a gallon 
and 25% ad valorem in gold; other charges on it amounted to 
$50.25 in currency. Reckoning gold at 135, what must he sell 
his wine for per gallon, in currency, to make 80%? Ans. $5.35. 


40. Two globes of equal density have a diameter respectively 
of 1 inch and 1 foot. The smaller weighs 14 ounces; what is the 


weight of the larger? Ans. 162 Ib, 

41. F, G, and H, enter into partnership for 1 year from Jan. 1. 
F furnishes $25000 for the whole time; G, twice that amount for 
9 months. H puts in $10000, Jan. 1; $5000 more, March 1; 
$15000 more, May 13 and Oct. 1, withdraws $10000. Their profits 
are $15500 3 how must this sum be divided; G being allowed $350 
for extra services ? Ans. F, $4500; G, $7100; H, $3900. 

42, A farmer wishes to lay off a rectangular garden containing 
2 acres, with a front of 200 feet on a certain road; how deep 
must he make it ? Ans. 485 ft. 7.2 in. 

43, A pound weighed by a certain grocer’s false balance makes 
1 lb. 1 oz. How much does he dishonestly make by selling 34 of 
his pounds of butter, if-butter is werth 50 c. a lb? Ans, $1. 


ir 


334 MISCELLANEOUS EXAMPLES. 


44, The rate of tax in a certain town is.008 and $1 for each 
poll. B has to pay for 5 polls; also on real estate valued at 
$5450, and $2250 personal property. He gets a note for $450, 
which will mature in 63 days, discounted at a bank for 6% After 
paying his tax from the proceeds, how much has he left ? 

45. To realize a profit of 154%, what price per pound must a 
grocer charge for a mixture of three kinds of sugar, costing re- 
spectively 10, 11, and 12 cents a Ib., there being 3 1b. of the first 
kind to 2 of the second, and 1 of the third ? Ans, 1244; c. 

46. P owes Q $3500 payable in 6 months, but pays half the 
debt at the end of 2 months, and $500 more 1 month afterward. 
How long after the 6 months may P equitably defer paying the 
balance? Ans. 6 mo. 24 days. 

47. Two persons start from the same point, and go the one 
due east, the other due south. The first goes twice as far as the 
second, and walks 20 miles aday. How far apart are they at the 
end of 6 days? Ans. 184.164 miles+. 

48, A certain quantity of water is discharged by a two-inch 
pipe in 4 hours; how long will it take 4 one-inch pipes to dis- 
charge 5 times that quantity ? Ans. 20 h. 

49. A and B erect and furnish a hotel. A contributes 4 lots 
of land worth $375 each, and $21400 cash. B puts in $5600 cash, 
and furniture worth $9250. They rent the premises for $7550 a 
year; how much should each receive? Ans. A, $4580; B, $2970. 

50. What price per yard in Federal Money is equivalent to 5 
frances for a metre ? Ans. 85 c.+ 

51. A owns # of a ship which is worth $68000. If I buy 2 of 
A’s share at this rate, for what sum must I draw a sixty-day note 
that I may pay A witltthe proceeds of it, when discounted by a 
broker at 7 per cent. ? Ans. $84421.67. 

52. A merchant owing 2500 francs in Paris, invests $5000 gold 
in a bill on that city, exchange being 5 francs 25 centimes to $1. 
Ife remits it to a friend, and asks him to pay the debt with in- 
terest for 6 mo., at 4%, and invest the balance in silk. 29624 yd. 
_ Of silk were bought; what was the price per yd. ? Ans. 8 fr. 
53. A grocer wishes to make a mixture of 280 Ib. of coffee, 


MISCELLANEOUS EXAMPLES. 300 


worth 26 c. a lb., out of four kinds worth respectively 23, 25, 28, 
and 30 cents a lb. How much of each kind must he take ? 

54. Paid, June 1, $1 for getting a note of $100 discounted at 
6%. When did said note mature ? Ans. July 31. 

55. Having 5 gal. of alcohol, worth $4.50 a gallon, a person 
wishes to mix it with two other kinds worth $4.25 and $4 a gal- 
Jon, and with water, so as to make the mixture worth $3.75 a 
gallon. What quantities must he take ? 

Ans. 5 gal. of each kind of alcohol to 2 of water. 

56. What price per pound avoir. in Federal Money is equiva- 
lent to 5 francs 50 centimes per kilogram ? Ans. $.4644. 

57. A grocer sold flour which cost $6 a barrel for $7.80. 
When this flour rose to $9 a barrel, what did he have to sell it for 
to make the same per cent. ? 

58. How many cords, and how many decasteres, in 3 piles of 
wood, each 4 ft. wide and 8 ft. high, the first being 10 ft. long, 
the second 15 ft. long, the third 20 ft. long? 

Last ans. 4.0775 decasteres+. 

59. According to the Connecticut rule, what was due on the 
note presented in Example 6, page 228 ? Ans. $1201.059. 

60. From a sphere of copper 14 ft. in diameter, what length 
of wire ;1, of an inch in diameter can be drawn, 4% being allowed 
for waste? Ans. d-mig7 far. 5-rde fh.6-im 

61. At $4 a rod, what will be the expense of fencing twelve 
acres of land in the form of a circle? In the form of a square? 
In the form of a rectangle whose length is three times its breadth? 

Ans. $621.98+ ; $701.07+ ; $809.54. 

62. I can buy 500 barrels of flour for $11 a barrel cash, or 
£11.20 on 3 months’ credit. Would it be better for me to buy on. 
credit, or to borrow the money at 6 4% and pay cash ? 

63. A father divides a sum of money between his children in 
such 2 way that the Ist and 2d together have $500, the 2d and 
8d $700, the 8d and 4th $500, the Ist and 4th $300, the Ist and 
8d $600. Find the sum divided and the share of each. 

Sum divided, $1000. 
= Ist receives $200; 2d, $800; 38d, $400; 4th, $100. 


i= 


TABLES OF MONEYS, WEIGHTS, MEASURES, &c, 


Federal Money. 
10 mills, 1 cent. 
10 cents, 1 dime. 
10 dimes, 1 dollar. 
10 dollars, 1 eagle. 


Apothecaries’ 
Weight. 

20 grains, 1scruple, 5). 
8 scruples, 1dram, 3. 
Sdrams, lounce, 3%. 

12 ounces, 1 pound, tb. 


Linear Measure. 


12 inches, 1 foot. 
3 feet, 1 yard. 
dh yards, 1 rod. 
40 rods, 1 furlong. 


8 furlongs, 1 mile. 


Square Measure. 


144 sq. inches, 1 sq. ft. 
9 sq.feet, 1sq. yd. 
304 sg. yards, 1 sq. rd. 


40 sq.rods, 1 rood. 
4 roods, 1 acre. 
640 acres, 1 sq. mi. 


Beer Measure. 

2 pints, 1 quart. 

4 quarts, 1 gallon. 
86 gallons, 1 barrel. 
1} barrels, 1 hogshead., 


1 Beer Gallon = 282 cu. in. 
1 Wine Gallon = 281 cu. in. 


Sterling Money. 
4 farthings, 1 penny. 
12 pence, 1 shilling. 
20 shillings, 1 pound. 
21 shillings, 1 guinea. 


Avoirdupois Weight. 
16 drams, 1 ounce. 
16 ounces, 1 pound. 
25 pounds, 1 quarter. 


4 quarters, 


1 hundred-wt. 


20 hundred-wt., 1 ton. 


Cloth Measure. 
24 inches, 1 nail. 
nails, 1 quarter. 


4 
4 
3 
5 
6 


quarters, 1 yard. 
quarters, 1 Ell Flemish, 
quarters, 1 Ell English. 
quarters, 1 Ell French. 


Cubic Measure. 


1728 cubie inches, plies its 
27 cubic fect, 1 cu. yd. 
40 cu. ft. of round, or 1 toa, 
50 cu. ft. hewn timber, 

16 cubic feet, 1 cd. ft. 
8 cord-feet, 1 cord. 


Dry Measure. 


2 pints, 1 quart. 
8 quarts, 1 peck. 
4 pecks, 1 bushel. 


86 bushels, 1 chaldron, 


1 Small Measure = 2 quarts. 
1 Bushel = 2150.42 eu. in. 


Troy Weight. 


24 grains, 1 pennywt. 
20 pennywts., 1 ounce. 
12 ounces, 1 pound. 


Miscellaneous. 
14 Ib., 1stone (iron, lead). 
100 Ib., I quintal. 
100 Jb., 1 cask of raisins. 
196 lb., 1 barrel of flour. 
200 Ib., 1 bar. beef, pork. 


Surveyors’ Measure. 
7.92 inches, 1 link. 
100 ~links, 1 chain. 
80 chains, 1 mile. 


10 sq. chains, 1 acre. 
¢40 acres,... 1 sq. mile. 


Liquid Measure. 
4 gills, 1 pint. 

2 pints, 1 quart. 
4 quarts, 1 galion. | 
814 gallons, 1 barrel. 

2 barrels, 1 hogshd. 

2 hhd., 1 pipe. 

2 pipes, 1 tun. 


Time. 
60 seconds, 1 minute. 
60 minutes, 1 hour. 


24 hours, 1 day. 
7 days, 1 week. 
365 days, 1 year. 
366 days, 1 leap year. 
100 years, 1 century, 


Circular Measure. 
60 seconds (), 1 minute, ’ 
60 minutes, 1 degree, ° 
80 degrees, sign, S. 
12 signs, 1 circle, C. 


[ 336 ] 


* Paper. 
24 shects, 1 quire. 
20 quires, 1 ream. 
2reams, 1 bundle. 


5 bundles, 1 bale. 


Collections of Units. 


12 units, 1 dozen. 
12 dozen, 1 gross. 
12 gross, 1 great gross, 
20 units, 1 score. 


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